I think paralysis by analysis is another way of saying you're working on the wrong things. If the analysis is good -- simple, explained well, nailing the problem and the fix -- then you're good to go. But people see a video camera out, maybe a training aid, and the golfer sucking it up and playing bad. They conclude "paralysis by analysis. Better to keep it simple." But the reality is they're working on the wrong stuff. Because what does "keep it simple" mean if the simple thought or idea in the golfer's head isn't their priority? They'll get worse going that route too.
Nail the biggest problem and how to fix it, and analysis/keeping it simple will end up working, no matter how one decides to look at the process. Just my two cents.
I guess in the end it all depends on how good the player is here we're talking about. Tour pro level, just believing in yourself and using one or no swing thoughts is probably not such a bad idea because they're already amazing.