So, the question really becomes whether a disability (in whatever form you choose to define that) is part of your "intrinsic identity", or something that is simply bound to your physical form? I think that's a pretty personal question that would have a lot of passionate responses.
I like this interpretation, and I think the Brandon Sanderson system mentioned earlier (i.e. the answer to that personal question is made by the individual being healed and/or changed - so whether healing magic erases a PC's scars is up to the player, for example) lends itself well to taking the answer to that question out of the hands of specific wording in the books.
Though that wording could easily have been put into the PHB or the DMG.
Of course, it's best to ignore recent RAW for a flavor question about this, IMO - a
lot of these effects have been pushed towards "the PC's character sheet is the shape of their body
and the 'shape' of their soul/'intrinsic identity' in all scenarios" in 5.5e if you go by RAW, for... dubious 'balance' reasons. Giving TTRPG spells video game ability balance patches has never sat well with me, especially for spells that are mechanically deeper than numerical changes. Competitive game/video game balancing very often loses the flavor of abilities over time because flavorful design is not the priority of a balance team. Flavor being secondary to 'balance' in one of the only realms - TTRPGs - where flavor can always be a priority because of DM rulings is something I've never liked about Crawford's design philosophy for rule writing. DM rulings as a crutch to add back removed flavor is not an appealing design approach to me, especially when the resulting balance is dubious at best and still 'broken' at worst. Mearls disagreed with Crawford about True Polymorph, at least around the time it was errata'd, but RAW now reflects Crawford's opinion on it (which differs so explicitly from the text in the 2014 PHB they had to change it multiple times - I'm fairly certain it was first changed to, "the transformation lasts until dispelled," before they eventually replaced "transformation" with "spell.")
They tried to do a similar sort of "not a real transformation" thing with Wild Shape in 2024, and walked it back because people really disliked it, but it wasn't the only thing. Magic Jar is like an extremely convoluted Disguise Self with some physical ability score bonuses (that also requires stealing someone's body and trapping their soul) now. Of course, it retains the potentially instakill vulnerability to Dispel Magic or Antimagic Field that it used to have as balancing, but with little more than a bigger HP pool as the benefit of the spell. Its flavor in 2024 RAW is downright incomprehensible.
Perhaps there is a middle ground here, where something like True Polymorph only remains a spell if the subject doesn't consider their resulting form their 'true' form, so people turned into newts can get better. Unless, of course, they consider themselves a newt now, in which case they already are 'healthy,' so there's nothing for something like Dispel Magic to dispel (or, if the newtification was a 'curse,' even Greater Restoration). This also saves the objects turned into creatures by True Polymorph from being outright killed by Dispel Magic, and may even let stuff like Resurrection spells work on them - and for them to have a consciousness that isn't implied to be eradicated completely by Dispel Magic (because you can't restore the original polymorphed form, even with a new polymorph spell - it'd be a new creature made from the same object).
The way I personally would handle it is, in essence, that a being's 'intrinsic identity' can change with time (but is not forced to), and what effects magic has in regards to that 'intrinsic identity' change - or even vanish altogether - in response. If a disability is considered not part of the 'intrinsic identity' by the target of a healing spell, the healing spell does its best to remedy that where it can - and at higher levels, 'where it can' might mean pretty much any physical ailment. Conversely, if a being's 'intrinsic identity' becomes exactly the form they currently possess... something like healing magic has nothing to change, and even if the form came from a polymorph originally, the polymorph spell is no longer actually making any changes to the underlying being, and thus ceases to exist as a spell on said being. I might add an additional duration to something like True Polymorph - like, say, having the 'intrinsic identity' adjust to the new form over like a month to a year if willing to do so - and have it become the true form of the being afterwards, requiring another True Polymorph to change (and that, potentially, the being in question might never allow the second polymorph to become its 'true' form at all, regardless of time spent polymorphed).
I don't see much real reason to say no to the question the thread presents. If you've got healing magic that replaces surgery, let it do that - hell, even in grittier settings this still works, it might just require a combination of actual surgery and healing magic. The idea that a removed, disability-causing tumor or something might just grow back after it's cut out and healing magic is cast, against the will of both the caster and the target, feels ridiculous to me unless the condition is caused by some magical ailment that healing magic isn't the right tool to deal with. For healing magic to 'remember' the tumor is there on its own, it would need something to go off of... which takes us right back to 'intrinsic identity,' and requires either that the one with the tumor considers the tumor a part of their own body or that the universe at large has decided that the immutable physical identity of the individual with the tumor is "individual with the tumor." I really dislike going the second route for a number of reasons, not the least of which is that it's almost invariably inconsistent with itself (i.e. the universe only ever decides that people
stay disabled -
becoming disabled in such a setting is almost always still on the table, and has no such magical resistance to alteration... until you're trying to remedy it).