I bet the broad idea of this theory has been discussed before, but Timeless Child does work with Regeneration Limits.
You've got this being who can infinitely regenerate that you're experimenting on to create the Time Lords
You've created an effective way to wipe their memory and force regenerations
Through the study of this process you learn how to cap regenerations for all time lords, grant new regeneration cycles if necessary, and create methods of stealing regenerations (though the stealing could just be a consequence of an exploited oversight, a bug in the system so-to-speak).
Once you've sufficiently got all of this created you can cap The Doctor, and all Time Lord's regenerations.
But oh fuck, she's gone renegade and now you've made her a fugitive.
Once you catch her, wipe memory again, wipe most, if not all Time Lords' memories of all of these shenanigans, and huck em all in the academy.
This situation could be part of the reason the Time Lords banned intervention, because of all the splintered timelines that, thanks to inserting the mechanism for regenerations into their bodies, all have simultaneously concurrent memories of various time lord origins, conflicts, historical events, and have decided:
"Ya know, this shit's getting too timus wimus; we gotta just observe, we keep changing shit allll the time, including our own origins. Remember when we were made with Looms? Remember when we weren't sterile?"
"Of course I remember! We're cursed with memories of all splintered timelines dude! We gotta toss all these shits in the untempered schism."
"Great idea! And of course nobody would ever make a time lord look into the schism as part of this new academy we want to set up now that we have time lord creation, regeneration capping, and memory wiping finished and perfected."
"Absolutely not! that'd be mad! Could you imagine? with all the disparate timeline memories we've extracted from everyone, and soon ourselves, that we've jammed in there? Why... it'd make a timelord go mad! We have to maintain our supremacy over all of time, and over evidently what are other Gallifreyans, who are explicitly not Time Lords, just people who live here."
"For sure. and also, if we make The Doctor look into the untempered schism, could you imagine how fucked we'd all be if he remembers all of their past lives? even as just a jumbled cacophonic mess? He'd go all maverick for sure!"
"Oh god, oh fuck, plus, what if he ever finds out he's the timeless child? Also, side note, maverick kinda blows... renegade is a better word, I think."
"Renegade is definitely better, Renegade Time Lord has a great ring to it, phonetically!"
"Thanks Omega, I'm glad we could sort these details out."
"Anytime Morbius. But man... it sure would suck if you and I went insane. These precautions might be thrown to the wind."
"Poppycock! My friend, we're the most sane time lords! We'd never end up in the antimatter universe, or try to destroy the world in Amsterdam, or be thrown off a cliff by the Sisterhood of Karn after becoming a brain, or be slightly cloned from Straxus, or become a giant skeleton"
"All of that's absurd, Morbius! We smoke weed in Amsterdam, why would one of us try to destroy the world there?"
"Exactly best buddy. Let's set a Denny's Date with Rassilon and Tecteun."
"Do we invite 'The Other' ?"
"Nah, dude keep talking about being a wizard and playing with spoons"
"Fuck. almost makes me want to undergo the time-lord memory removal process even quicker. Can we merge a timelord with another timelord? While we're memory wiping we might as well get that weirdo reigned in too, merge that dude's entire being with The Doctor."
"Agreed... plus that would let us replace the regenerations we burned through after we put the 13 regeneration cap on them but forgot because we went to Amserdam while and sent them on a few more missions... We should invite Artron to Denny's though."
"Oh fuck yeah, Artron rocks. Make sure not to accidentally make a wish around him though."