Later on in the story, Solon explains to Morbius that the reason he became obsessed with using the Doctor's head in the project is because, as a Time Lord, it would be more appropriate - he says he likes the idea of having a representative of the culture that ended Morbius's life being the literal figurehead for Morbius's reign. But Solon's reaction to the Doctor's head happened before he found out that the Doctor was a Time Lord, not after - when he opens the door, he's eagerly anticipating 'human' visitors and he only realises what the Doctor is after the Doctor starts chatting about all the different heads he's had.
The order in which Solon finds out the Doctor is a Time Lord and in which he falls in love with the Doctor's face changes the entire story. It goes from being a story about a scientist seeking appropriate organs for his work, to being a story about a scientist who wants to use Tom Baker in a thing he's doing because he's the most gorgeous man he's ever seen. There's been stories before this where characters found the Doctor handsome (Polly, seeing the Doctor after his makeover in "The Macra Terror") or dashing (Liz giggling about the Doctor flirting with her in "Spearhead from Space") but never before have we been presented with a character who unambiguously finds the Doctor beautiful.
Of course, this Doctor isn't one of the most unambiguously beautiful ones. And the things about him that are commonly admired don't get any focus here - his sparkle and his smile get little attention and his famous voice goes without comment, save for a scene where Solon isn't present which places great emphasis on the Doctor saying the word 'Popocatepetl' simply for the sheer pleasure of hearing Tom Baker say the word 'Popocatepetl'. In fact, he spends much of the story unconscious. But his expression doesn't even appear pained or vacant in most of his unconsciousness scenes - instead he wears a proud, beatific expression like a carved knight dreaming on his own coffin. The charitable description of this Doctor's face is 'Classical', and with his eyes and mouth closed that image takes over - scenes in which he camouflages himself amongst Solon's Classical-styled sculptures do a lot to emphasise this way of viewing him. In fact, the more common and less charitable description is outright rejected - the giant-toothed, swivel-eyed Mutt head and Morbius's protruding eyes are there only to be images that the Doctor is not, so inferior as to be beneath consideration. It would be easy to portray Solon's opinions of the Doctor as laughable - much of the rest of the story portrays Solon as laughable - but instead the story draws us into his way of thinking by showing it to us. The story distances us from Solon by making him ridiculous, and yet pulls us into him when we end up sharing the same visual pleasure, and the effect is a deliciously creepy sense of guilt by association.
For it's really unusual just how much we end up relating to Solon in the story considering he's both the antagonist and an almost comically horrible person. The slightly absurdist tone is part of what makes him as relateable as he is - he's fallible, makes obvious mistakes, driven by emotion, and we can all understand that because we've all been that person. While he's subordinate to Sarah in terms of an audience insert, losing his viewpoint status when she is present, he is superior to the Doctor, who is only used as viewpoint in his altercations with the Sisterhood. This Doctor is one of the most relateable Doctors, the Fourth Wall Doctor who didn't need a companion and derailed his own show by becoming his own audience, and yet he is presented in this story mostly as an object to view, with the exception of when he's facing a group of women defined by their inability to sense him.
And yet even the Sisterhood, who can't sense the Doctor and Morbius because of their Time Lord minds, can scry for him. With the effort of her Sisters, Maren is able to see an image of him, viewing him on a screen much like the audience is doing through her perspective. And the vision she sees of him is a shot of his head.
The very first thing that we see in "The Brain of Morbius" is a stylised image of the TARDIS emerging from a whirling psychedelic pattern, which resolves into an image of the Doctor's head. He is wearing a blank expression, and his face is lit with symmetric lights that make his features seem inaccessible, reducing them to geometric shapes like the light show he emerged from. The basic truism goes - introducing a character with a body part makes the audience see them as an object; introducing a character with a shot of their face makes the audience relate to them as a character with feelings. But the Doctor's head is presented as just one object in a list of objects. This is the TARDIS; this is the logo; this is the disembodied head of the Doctor.
The Hinchcliffe era, the era The Brain of Morbius belongs to, comes at a time in Doctor Who history where it's beginning to self-mythologise. Regeneration of the Doctor has happened three times (or, at least as far as The Brain of Morbius is concerned, that the audience knows about); like the Rule of Three in folk tales, three times is enough times for it to develop from being a shock twist to an ongoing pattern, a feature of the show, a selling point. One of the earliest scenes in the story is set up to remind us of this - immediately after the scene where Solon falls in love with the Doctor's head, the Doctor does some of his trademark Fourth-walling when he reminds us that 'a lot of people seemed to like' his last head, referring to its success with fans as much as with Jo Grant and Benton. Except he doesn't actually use the word 'head' when talking about it - he refers to both his previous head and his current head as 'models', as if he's merely changed his car.
When the show itself treats the head of the Doctor as both instantly striking and fragmentable, the physical centre of the show and focus of its popularity - when the Doctor himself considers his head to be an interchangeable device for carrying his brain around in - when Solon is the audience insert character for much of the story - why do we still make the old fanboy jokes about why Solon didn't just transplant Morbius's brain into the Doctor's whole body?
It's worth emphasising just how little interest Solon actually has in the Doctor's body. In theory, Solon plans to violate it, and the gothic horror rape allusions used in various previous Tom Baker stories (a soldier sadistically torturing a female POW by forcing her to hallucinate a snake crawling up her leg?) could easily appear here if that was the story being told. Solon's given more than enough opportunity to creepily touch the Doctor's lips or lick him and or whatever - he certainly would have done in a lesser story, which would have used it to cheaply indicate that we are supposed to hate him for being depraved - but the closest he comes is running fingers over his chin at arm's length with the expression of a man who can't quite believe he's allowed to touch the exhibit at the museum. Instead his appreciation of the Doctor is based around Solon's aestheticism, the part of him that makes sculpture and talks about letting wine breathe. When surgically examining the Doctor in the first episode, the tools he uses aren't medical-fetish/phobia shiny metal; he listens to the Doctor's heart with a gorgeous turned wooden stethoscope, and when he takes the scalpel its carved mother-of-pearl handle glints in his heavily ringed hands before he runs that end very gently over the Doctor's throat, holding it like you would a feather rather than a knife. The tools might be intended eventually to invade but they never do, instead passing over the Doctor like a gaze. Solon's obsession with performing the operation perfectly is a means of honouring the person he wants to put inside the superb head - if Solon hadn't cared so much about decapitating a man in a manner he thought beautiful, the Doctor's headless corpse would be stinking up the laboratory before the end of the first episode. Solon is less a mad scientist, and more a mad artist.
The only character from where body desire appears is Morbius - furious about his own lost looks, dysphoric about his current body, and far too responsive to the Doctor's insults. The Doctor describes the mindbending contest as 'Time Lord wrestling', which has certain connotations considering the Greek sculptures in the set design that you wouldn't get if he'd called it 'Time Lord arm-wrestling'. But even when Morbius overpowers the Doctor and starts crowing about how he's violating him, all he appears to be doing from the audience's perspective is observing the images of his heads - and at the same time we are shown that he is looking at parts of the Doctor so secret and personal that not even we know about them, that (played as they are by production members) lurk slightly on the wrong side of the fourth wall.
This Doctor is one of the most active Doctors, the one famous for bending stories around his will to the point of derailing a show that was supposed to be about him anyway. And yet his only option in the fight against Morbius is to submit to his gaze, to allow his self to be destroyed through the viewing of his head.
But if the actors who played the Morbius Doctors are relevant, it has to be relevant that it's a Tom Baker story - the actor who most used the Doctor's face as a means of allowing his self to be lost. The scene where the Doctor has been drugged but first appears to be just drunk is clearly playing with actor personality rather than Doctor (the precedent, the Doctor's enthusiasm about the pub in The Android Invasion, was a Baker rewrite) - so if there's already one biographical idea in there, maybe all this face-peering draws from Baker's own quirk. Baker suggests it developed as a result of spending years in an environment where he was religiously forbidden from looking at the faces of others in the name of modesty; he has written about falling in love with beautiful male fingers and necks, losing perception of other bodies as anything more than individual parts, becoming desperately and immediately infatuated with another novice simply because he inadvertently looked up and saw his magnificent head. Maybe that's why Solon, living in a secluded, all-male cathedral-like environment as part of his religious devotion, was so struck by the Doctor; piecing together ideal bodies from disconnected limbs, he was able to finally complete the image with a face, and struggled and failed to put his faceless God in it.
No wonder Morbius tells Solon that he doesn't want the face of a Time Lord, and to instead forego getting a face altogether. (I love how much of an unhappy, miserable, jealous family the three castle-dwellers are.) In a setting where all characters are seeking physical completion and stability - even the castle he lives inhas ball-and-socket buttresses, suggesting transience - he chooses to remain incomplete, foregoing an organ that allows others to see his thoughts in favour of one that shows only his brain. Those aren't his faces during the mindbending, and in fact the only time we do see his face it's when the drunk-drugged Doctor finally recognises the statue - which causes the Doctor's dream in which he feels a direct connection to Morbius's thoughts. So Morbius fails because he refuses to submit to allow himself to be seen. Notice how Sarah shows no fear of Morbius while blinded, treating him as an equal and even a potential ally, and only screams when her vision comes back and she sees that he still cannot be seen! It's appropriate that the faceless Morbius is damaged by 'static buildup', too - robbed of a face to allow him to express externally, his thoughts causes energy to build up internally until it damages him.
So even if Solon's fixation on the Doctor's head and Morbius's fixation on the Doctor's faces is objectifying, it doesn't mean our gazing at him is. After all, even if this Fourth-wall Doctor knows where the camera is, he still can't see us watching him at home. Our gaze at him isn't us communicating with him - it's allowing him to communicate with us. Without us watching none of his adventures would happen, because he is performing for us, so it's because we watch the Doctor's face that the Doctor has an identity at all. Without the immodesty of a face, he would sink into the depersonalised introspection of Morbius.