My pet theory on Putin's ability to maintain power and avoid a coup at this point is that he has spent the last 20 years rigging this system (or something similar) to a personal dead man switch, and is actively holding the entire country of Russia hostage. Everyone in his inner circle is made aware of it once they get close enough, and any leak about it ends with you falling from a window. He's just cynical enough that it goes from "comic-book farce" to "terrifying possible reality" the more you think about it.
Kinda sorta. My understanding is that he maintains a situation where everyone in power is dirty and you must be demonstrably dirty to get close to the levers of power. That shared guilt helps keep the ruling clique together. Add to that the very real threat of assassination, and it seems pretty airtight.
Your idea has the flaw that the leak would be damaging if done, and sometimes people are willing to take action at suicidal personal cost.