Perhaps I'm wrong but my understanding is that mirror adhesive is designed to help support a flat mirror, not a medicine cabinet. The weight per square inch of glue surface is vastly different between the two, and the physics of the load applied are very much different. That unit should have been screwed solid to a stud. Even if they used a construction adhesive designed for the job, the tiles are not designed to support that weight and can pull off the wall since it would be the next weak point. If they glued it to drywall then I would expect the paper to rip off since it's only glued to the surface....
How are you going to replace it when the lights in the doors fail or a mirror breaks? They've turned a simple cabinet replacement project into another bathroom renovation since you will now be having to rip it off the wall and most likely having to be replacing tiles after you do that, I hope you have spare matching tiles for when that time comes. Does his guarantee cover that added expense also?
It bugs me when people cut corners because they can't be bothered to get the proper tool (that any hardware store would have carried). He saved himself the cost of a $8.00 drill bit but cost you a huge expense and time when you go to replace it.
I would invite him back on his own time to install it as the instruction manual specifies, but for some reason I don't think he'll show.