Even though WWE is a publicly traded company, Vince McMahon owns somewhere in excess of 60% of all WWE stock and the McMahons have roughly 95% of the voting power within the company. So yes, while he does own explanations to stockholders and has to keep them apprised of the goings on, said stockholders can't exactly force Vince to do anything as far as hiring people, firing people, changing strategies, etc.
The only way in which Vince loses power or is no longer the majority shareholder would be if someone ponied up the dough to buy more stock than Vince himself. It'd take some serious money to do that. Even after taking such a huge hit in the past few days, WWE stock is still worth somewhere in the vicinity of $1 billion. Even if a major heavyweight like Ted Turner or Donald Trump wanted to oust Vince, that's still a helluva lot of money even for them. Turner's worth about $2.2 billion while Trump is worth $3.9, so while it's theoretically possible for them to swoop in and buy up WWE stock like it's candy to such a degree that they own more than Vince, it's too much of a risk. Roughly a quarter of Trump's wealth and nearly half of Turner's would have to be invested and that's just not gonna happen.
Being a publicly traded or privately owned company has had little to do with WWE's change of direction over the past decade or so. For instance, WWE was still TV-14 and a publicly traded company well into the 2000s. They were still doing some pretty raunchy stuff during the 2000s but, all in all, Vince is very much a PG oriented type of guy. The Attitude Era came about all in all as almost an act of desperation on Vince's part as it was such a radical departure from the more family oriented wrestling that was not only common when I was growing up but is also common today. For every CZW, there's a dozen other promotions that operate under the traditional format of wrestling being family friendly. As to why Vince took WWE public, I don't know really. When it comes to financial analysis on a corporate level, I'm just not all that well up to speed.
I don't look for WWE to go back to being privately owned. While the stock sinking as it did is certainly a big disappointment for them and a big loss, it's still not the end of the world. Stocks skyrocket and plummet all the time. If you ever flip it over to CNN for any length of time and see the numbers for the Dow Jones, S&P 500, etc. down in the lower right hand corner of the screen, those numbers are all over the place. The stock market is like being led blindfolded onto a roller coaster; you can't see where all the dips, rises and loops are, so it's damn hard to prepare for them because you don't know when they'll pop up. For instance, you may tune in this Monday and see the Dow Jones gain 150 points while it might drop 200 points by Tuesday, gain another 200 points on Wednesday & Thursday combined before plummeting another 200+ points on Friday.