You are correct, he wasn't the director. But, given that (in addition to being the lead actor) he was also involved with the story of the film, was one of the producers and described the film as a passion project of his Baldwin was incredibly heavily involved in it.

Was he really supposed to check it over to make sure the diversity hire could do her job properly?

Given that there had been several gun safety issues previously on the set, yes. Given that some reports (don't know if they are confirmed or not) say that crew were using that gun - and others - for shooting cans during lunch right before the shooting, yes. Given that they were using a gun capable of actually firing, yes.

Oh, and he could also have just followed gun safety 101, which includes things such as "never point the gun at someone unless you're willing to shoot him", "always check your backstop" and "don't pull the damn trigger unless you're comfortable destroying whatever the gun is pointing at"

I'll agree that, by all accounts, the armorer was sloppy if not incompetent. Does she deserve some of the blame? Quite possibly. But, even if the live round was entirely her screw-up, what Baldwin did should never have been done. There is ZERO reason to ever point a gun at someone's chest and pull the trigger unless you want that person dead.