Except for the beyond exceptional Improv artist, a pure improv game won't be cohesive or make sense. It will just be a bunch of non connected encounters and events.
And random tables don't really work as "prompts to the imagination", Unless you make massively complex tables. And this goes right back to massive prep time.
Illusion smooths everything out to make the game work.
Well, first of all, at no point in this clarification did you actually answer my question. You made no appeals here to the need to deceive the players about the nature of the game that they were playing or to give them a false choice. I'm willing to except pretty much any clear definition of Illusionism however you would like to state it, but it's not really clear from your response that you have a definition of Illusionism at all. Your statements work equally well and perhaps better if you are talking about improvisation or imagination. They miss what I think is the key idea people are trying to capture when they talk about "Illusionism".
As for the three clarifications, while it's true skill with improvisation is required to make a cohesive and non-contradictory setting, Illusionism itself doesn't make that easier. One of the problems with illusionism that allows it to be detected by the players is that if the DM is inventing things on the spot all the time, it's hard to keep the story straight. Even if you have everything written down, it's hard sometimes to remember how all the things should impact a scene. But it's much worse when you are inventing and improvising (whether Illusionism or not). For example, you may forget that the elf was owed a chance to automatically detect a secret door or that 30 minutes a go the party flicked a wand of secret door detection and create a secret door whose existence should have had an impact on the game earlier.
I think lots of people do find that random tables do work as "prompts to the imagination" so your assertion seems like it needs explanation. And in any event it's not clear how Illusionism will successfully fill any gap in the GM's imaginative abilities if a random table isn't capable of doing so.
The purpose of Illusionism is to fix the drama so that the GM gets the story they want regardless of what the players do. That can either be to smooth the way forward when obstacles might otherwise get in the way of a particular story, or else it might be put obstacles up that steer the players back onto the desired story when they start doing things that are unexpected. An example of Illusionism might be if early in a story the PC's decide they are going to murder the NPC that you intended to be the BBEG and you decide to fix this in someway so that it doesn't change the story either because the NPC ought to be able to easily kill the PC's at this point in the story or because the PC's are unwantedly successful in their murder plot, so you decide on the spot that the NPC is actually a simulacrum of the real NPC and chuckle to yourself that now that the PC's think that that NPC is dead they'll never suspect that he's the evil mastermind of events. What Illusionism fixes is anything going wrong with the story you the GM want to have happen. That story can be one you planned out, or the story you think the PC's have signaled that they want and you've decided is a good story so you'll validate it. Either way, Illusionism lets you decide what really happens.