OP has an interesting theory, as I was wondering the same thing. Yes it would be very illegal, but that hasn't stopped other CEO's from doing bad things in the name of profit (Dieselgate anyone? Enron? WorldCom?). But what other theory makes sense for why the CEO would do things every few weeks everyone knows will send the stock down, at least temporarily? He's certainly smart enough to know by now what these acts / tweets will do to the stock price. The fact that he keeps doing it sure makes it look like he's doing it on purpose for some reason. Or maybe he really has gone insane.
That would make for some wild reality show drama, like our White House, to discover he's part of the whole shorts-media conspiracy to make tons of money off the unsuspecting public with the stock swings...