I shot a small southern whitetail with a 300WSM at 35 yards, quartering shot, deer heading out at 2 oclock, got him slap in the pumphouse.
The load was a Nosler Ballistic Tip 150gr in a Federal factory load.
Yeah, it was WAY overkill, and in theory I should have blown the deer's innards all over the place.
However, what ACTUALLY happened was the deer was dead before it hit the ground, it's innards were turned to goo, and there was no exit wound.

I wanted to figure out why, so I started doing ballistics reaseach, even going so far as to directly ask Nosler to give me their best guess as to what the bullet did.

The information I gathered was pretty interesting.
Nosler figured that, since it was so close, and the bullet was going so fast, the bullet essentially exploded inside the animal.
Which, based on what I saw while gutting it, seemed to be reasonable.

Depending on the load, a 30-06 is zipping along in the 2600-2800 FPS range, based on what I read.
I personally like core-lokts because they're super cheap, the brass is pretty durable, and I can get 10-12 reloads from them.
I won't use them on animals, just because I like other things for that purpose.
My theory on why the jacket stripped off is that you hit something during the bullet's flight that changed it's trajectory as well as it's shape, which would account for both the deer taking off instead of being DRT, and also the jacket being found on the ground.
I'd look along the bullet's flight path, within 20' of where you found the jacket, and see if there's a small twig or something of that nature that's either blown off or nicked.
Again, this is my theory.
Seems plausible, at least based on what you wrote.
Either way, you have a nice pair of dead deer in the freezer.
Dead is dead. Only difference is how far you had to walk to collect your quarry.