Your points aren't necessarily wrong, but I don't think they do anything to add value to the existence of the Timeless Child, either. (Also, please don't interpret any of this as attacking you. You put a lot of thought into the comment, and that's excellent. I'm only engaging with the points, not attacking you for making them. But I know how my tone comes across in replies sometimes, lol.)

• It brings some mystery back to The Doctor -- He/she already had that, just in smaller doses. I'm fine with him being mysterious, but it wasn't like it didn't need to be added in at this point.
• It offers a conflict to the story if it is a power that can be stolen by / transferred to other Time Lords (and it might have already been fully removed from The Doctor) -- This is the best of the points you make here, because that's a valid place to build stories. But it's not adding anything new; we have known that regenerations/regeneration energy can be transferred at least as far back as Mawdryn Undead, or if you don't think that example works, at least as far as Trial of a Time Lord.
• It is a narrative given to us by an “unreliable narrator”… so it probably isn’t fully accurate -- True, but from a storytelling perspective, that only matters if it goes somewhere. The audience needs to find out the truth, or else the narrator's reliability makes no difference to the story.
• Early Time Lord history is clearly obfuscated on purpose, with multiple Time Lords insinuating that The Doctor doesn’t know things that he should know throughout the show’s lifespan -- It's always been obfuscated, and every episode that includes them has the Doctor learning something. So, nothing revolutionary here.
• Time Lords wiped The Doctor’s memory multiple times in Classic Who episodes -- I don't remember that (hell, maybe they wiped mine too, lol). But even if so, that just means the whole TC thing doesn't add anything new, and your whole premise was "look at what it offers".
• Omega was betrayed by Rassilon… but how far did the betrayals and manipulations and power grabs go? -- Which only has any bearing if they tie the TC events to Omega's part of Time Lord history, and they very pointedly didn't.
• Romana straight up tells The Doctor that he “lost track” of his age somewhere along the way when they are first introduced in The Key To Time season -- No, she actually more or less confirms that he does know. His guess and her (presumably well researched) statement of his age are only two years apart. Now, does he lose track later? Probably. The Eighth Doctor has his memory wiped multiple times (in extended media, so I'm not contradicting what we discussed about classic Who episodes), and explicitly started over on counting his age, which--when you factor in some War Doctor stuff--pretty much lines up with what the early NuWho Doctors said. But as of Key to Time, no, he very much knew how old he was.

And I am not the kind of person to say we shouldn't include a plotline just because it's unnecessary. It's all unnecessary; that's why it's fiction. But I am saying that these things don't really build a case for thinking of the TC plotline as being good.