In the comics (I think - I don't read them; I just looked a few things up), the trial is a result of Kaine murdering people. Which is presumably not the case here if KoD is desperately trying to keep Spider-Man alive, as that doesn't jive with Minish's clarification.
What I am about to say pertains to the source material and in no way necessarily reflects anything about the game that is based on it. You should not take any of this as facts about this game, it's just me being a Spider-Man nerd and talking about cool bits of lore.
In the comics, the 4 part story titled "The Trial of Peter Parker" was actually quite interesting because it featured not one, but
two trials of the titular character. Peter had been arrested for murders committed by Kaine (his clone) that had DNA linked back to him at the scene of the crime—pretty convincing evidence. Ben Reilly (his other clone) took his place in jail, pretending to be Peter, while Peter remained at large looking for answers. During this time Peter was kidnapped by Judas Traveller and also put on trial for his actions as Spider-Man, being prosecuted by his worst villains (like Carnage, for example), while he was being defended by Kaine, the murderous clone who got him in trouble in the first place. So while Spider-Man was on trial in Judas' court, Peter (Ben) was on trial in a New York court. It's actually a pretty neat story that came out of the otherwise mostly crap clone saga.
Again, this game might be based on those events but you should know by now "loosely inspired by" does not mean "follows very closely"... Don't take anything I have said here as facts about the game.