There are specific things that are the role of the top level executives, like certifying the company's SEC reports. But I'm not sure how having someone other than Musk independently and competently do those things would be expected to hurt anything.

Meanwhile, wouldn't it be completely normal and legitimate for a company to give the VP of Engineering or Operations a free hand in making engineering or operations decisions?

This. Tesla is a publicly traded company. It makes clear, legally-binding representations of who its officers are and what its organizational structure is. If Tesla represents that Musk is not longer a chairman, while in fact he is, then that's (again) investor fraud.