Yes, I looked at the photos closely. Look at those photos again, they are all extreme closeups of minor stuff. MAYBE the beads are out of whack, but there is nothing in those photos that show they are. Those shims, hackish as they may appear, show that the guy hanging it was at least working to get the geometry right. I've seen metal bead put on beautifully that couldn't be finished. You may not like staples but USG says crimp, nail, or staple: not drywall screws. None of the photos are pulled back far enough to see the screw pattern. So, maybe it wasn't secured properly, but there is no evidence.
I assumed the guy hanging was the apprentice, but the OP states he's a 30 year vet. I can see that as well. Look at those pictures again and picture a crusty 30 year vet getting the important stuff right and not caring at all about some dents in the drywall in between corner beads that will get filled with a 1/4" of mud. Or a 3/16" gap between boards that will get pre-filled. But spent time to get the bead in line. "Know the rules so you can break them" We don't know, He could also be an alcoholic on a hangover. But there is NOTHING in those photos that can't be dealt with in minutes by a quality guy. That doesn't mean there aren't flaws that we can't see, but all we have are those photos.
To the OP: In my experience either that mesh tape was always there, it was sanded down, the lighting or paint changed to reveal it or there is some magic in your house that i don't understand. I've seen mesh tape fail a bunch but i don't know how it could migrate through mud without cracking. If I'm wrong I'd love to learn something new.
In regards to this job, you've done what you can, you either need to fire them or trust them at this point. Like others have said, your best bet is to be nice to the guy who shows up to finish.
Merry Christmas to all and I hope it works out in the end.