I doubt Putin is anyone's useful idiot. I'm no expert, though.
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I'm pretty sure he is someone's useful idiot, but not Xi. Xi is a useful idiot too.
The real people behind all this "don't exist", or otherwise they'd be quite famous.
It's one of the laws in the 48 laws of power. Being in the ear of the visible power figures is more powerful than being the one who people might want to send to the guillotine. Being another step removed is probably even more powerful, and less risky.
That would be why Xi shows up for BRICS confab and Putin is on live feed... of course!
I'll just add that, in this game, the "wild card" is a valid game piece, and he's got levers too.
The only people in the world not being influenced are those doing the influence and the tiny sliver of the population who automatically look for the strings at every turn.
They are everywhere, and they are nowhere. But they can't control everyone. Some people have enough humanity left in them that they don't trigger the agent program, when you reveal things to them, even if they think you might be crazy.