Yay logistics!! Stoked to see someone else post about this, as it's actually something I've put way too much thought into. Imo, it only makes sense for the Empire to have a very organized and efficient system going back hundreds of years, and I honestly see the Korriban academy as an outlier in its chaos - k'lor'slugs georg, if you will.
(disclaimer that the following worldbuilding is all pulled out of my ass - idk what the devs' intentions were, but i've connected the dots they put down and drawn a few of my own, and this is the picture that makes sense to me. feel free to take bits for your own headcanons or leave it, as you will :)
I've gone with the idea that the chance of notable Force-sensitivity is roughly one in a million across the Imperial population (much higher in Sith lineages, but lower in the vast majority of families). In a planet with a population of 10 billion, that would be roughly 10,000 Sith + acolytes (in peacetime, when they aren't all dying - functionally it's less, even during SWTOR's cold war era. infighting go brrrr.) To deal with those thousands of acolytes, there are several government-run Sith Academies on each Imperial planet, one per major city, each capable of handling at least several hundred magical angry children. (This is most likely the Sith Inquisitors' biggest job, imo - enforcing Academy rules, weeding out dissenters and making sure the youngest acolytes are effectively indoctrinated to prevent things from boiling over once they hit puberty. Picture your worst teacher but with Force lightning and an interrogation chamber. Yeah.)
That said, some acolytes will be privately educated, especially at younger ages. Of course, some of those acolytes end up becoming militant cultists or fanatics of the houses or individual Sith who took them in - but it's still permitted by the Dark Council, as it eases strain on the public Academy system. Not to mention the Councilors certainly benefit from having their own private cohorts... and trying to put a stop to the practice would just cause a civil war even faster.
Canonically, Scourge and Malgus were said to have trained at a Sith Academy on Dromund Kaas, and the Sith had to study somewhere before Korriban was reclaimed. So I've expanded that to a more reasonable size: in my headcanon, there are dozens of Academies across Ziost, Dromund Kaas, Dromund Fels, Ashas Ree, Begeren, etc., each having a few hundred to a couple thousand students at most. And then there's Korriban, both the oldest and the most recently (re)established of them all.
At the time SWTOR begins, the Sith Academy on Korriban is far from the only one, but it's the most dangerous and the most prestigious; anyone who survives its trials is almost guaranteed an apprenticeship, as the Dark Council and other high-ranking lords hang around to pick off the most promising candidates for themselves. I figured it would actually have fewer acolytes than most Academies - usually a few hundred at a time, more of a village than a city - but who are constantly coming and going, rarely studying there for longer than they must to complete the trials (which are themselves definitely at least a few weeks/months, longer than the game would imply). Korriban has a very small long-term population, due to the risk of madness and hostile terrain which forces its inhabitants to rely on supplies from other worlds, so its overseers would have to take applications or travel offworld to scout acolytes from the wider Academy system.
As a result, the acolytes who end up on Korriban fall into two categories - the nepo baby/tryhard Imperials (SW, Eskella, Vemrin, Ffon, etc.) and the slaves and aliens (SI, Kory, Xalek, etc.) who were either born on the planet or sent there to die, as the overseers' unsubtle backlash to the changed admittance rules and fodder for their preferred acolytes. The majority of acolytes in the other Academies are more middle-of-the road, and don't have to deal with quite as much competition; their trials are less deadly, and no acolyte-on-acolyte killing is sanctioned, but they're also less likely to get apprenticeships and someday become lords or Darths.
To illustrate a couple examples of how this all works practically:
my SW was a freeborn zabrak whose Force-sensitivity was discovered early on; he was educated by the Sith lords his parents served until he was old enough to be sent to a public academy, where he survived the usual anti-alien hazing and was successfully indoctrinated; he would have graduated as a middle-of-the-road Sith and gone on to work as an assassin for his lords, if Tremel hadn't scouted him as a potential candidate for Baras' apprenticeship.
my Cathar SI was born and raised as an Imperial slave, but her Force-sensitivity being discovered as a violently rebellious teenager meant that she was sent straight to Korriban in hopes that the system (via Harkun) would get rid of her, or at least spit her out as somewhat controllable cannon fodder.
Thus, the game's origin stories end up as obvious endpoints of the dichotomy this system creates. (I was very amused when I realized this. Ah, serendipity.)
I love your take on the trials' different stages, and the quarters - the calendar system and timing is a part of this that has always escaped me. So far, I've figured that there's an eight-year "lower" program meant for kids 6-14 and a six-year "upper" program for 14-20; final trials for the 20-year-olds last several weeks and happen year-round on Korriban, with acolytes steadily coming in from offworld to compete for the lords' attention and hopefully graduate with all their limbs intact...
...but that's as far as my brain managed to get. Korriban apparently has a 28-hour day and a 780-day year (more than two galactic standard years), for whatever that's worth. (on the subject of numbers, it also has a gravity of 1.4, which exhausts me just thinking about 💀)
Also, the idea that some overseers or their flunkeys are sneaking into the tombs to reset traps and plant warblades is perfect - it's the kind of worldbuilding/lampshading that's both very funny and fitting, imo. 💖
