I think this comes up a lot, some people used to get upset when posters would describe programs as "concentration camps", claiming it minimizes what happened to jews. Other people claim gulag is also too "loaded" of a term, since most people associate it with communist russia, and by saying it (according to their argument) people will be turned off, and stop listening.
I kind of think that that gulag is a good term though. But I think it's time that we invented our own term. I usually use private prison because that's really how I felt it was. The only reason these places exist is because someone is funding them, in this case it's parents. But prisons are for people who broke the law and are being punished by serving time. Programs are not like that, since the kids were not convicted of crimes (and in most cases never committed any in the first place), and the amount of time spent in a program is very relative. It depends on how much money your parents have, how good of a bullshitter you are, the whims of staffers, etc. You don't get out after 1 year, so in that way it's not like prison at all. You can get out sooner, if you convince them you are a brainwashed mormon, or whatever the hell robotic non-existent stepford kid they are seeking. I still don't quite understand the demented dreams of parents and what their deluded expectations might be. But based on marketing, and the "tesimonials" of the freshly brainwashed, I'd say their expectations are bordering on lunacy, sickening at the very least.
So I am out of ideas. I would maybe suggest "coercive cult" but that doesn't convey the fact you are , indeed, in a prison with no rights. Prisoners actually have more rights. So I agree with the OP, I think gulag is the only term that comes close. It's time to come up with a good term that properly conveys the horror and intricacies of what the industry refers to as "programs". Maybe the word "program" in this context, will eventually morph into a very negative word. Maybe in a hundred years people in university will be writing about what is happening, and the word "program" to them, in the context of this time period will be extremely negative in the same way gulag or concrentration camp is to us. Maybe they will even have arguments about the proper use of "programs" in their present future, to not interfere with the true meaning of the word, a horrific history of child abuse. I'm rambling, but seriously. The reason I bumped this was because the OP posted at 4:20, and that can't be coincidence, it's a sign from the divine. one which i intend to heed.
