I just assumed that the whole 'war college' thing is a left-over from when Earth was not part of the Federation, post Burn. In Disco, we saw that Earth had it's own defense force and I'm assuming it is still in existence, in-effect right beside Starfleet, and maybe fodder for future integration back into Starfleet as things progress.
As to the show, the first episode was all right, but these last two were hokey nonsense! My gawd! That's not to say I have a problem with the plot, but just the decision to turn the Academy into basically a futuristic version of Animal House, and not done very well at that. Drop the attempts at comedy and get to the overall story! In fact, this would have been better if these cadets were third or fourth years, giving them more leeway to get away from the frat house um, the Academy.
May give it another episode before deciding to abandon this one. Man, really wish they'd given us another season of Prodigy...