In terms of gritting teeth Mayo isn't referring to the backswing and they probably aren't doing that on the downswing either. He's saying it to drive home his point that the swing isn't a "loose" or relaxed motion.
Trying doing anything where you create a lot of force and be loose, can't happen.
The grip is still firm on the backswing, if it wasn't you couldn't load the club upward.
Check out how these player's biceps and forearms look on the backswing, definitely not relaxed or loose.
Check out the "tension" (creases in the shirt) they create with their lats, shoulders, ribs to bend and stretch.
For me it's just a result of a good grip. The grip feels very secure in my hands, like I said earlier, someone couldn't yank the club out of the hands. But at the same time I'm not white knuckling it. It's firm but I'm not actively squeezing it.