"Do you guys just shoot at the animal hoping for a kill, but trying to wound it to slow it down..."
No. The idea of "Use enough gun" is that if you make some slight error in aiming, or if the animal moves during that 0.2 seconds between the time your brain says, "Shoot!" and your finger actually moves, the hit that's not on the exact point of aim will still put the animal down long enough for that second shot.
I don't care how skilled you are, Murphy never quits.
A standing-around deer generally seems to go down and stay down. Mostly. Once a deer has been hit and it's "only" a serious wound without being a devastating wound, the dadgummed things seem to take on the characteristics of Superdeer: They'll maybe soak up several hits before giving up. Adrenalin is the usual alleged cause.
I guess every deer I ever shot, I did so with the expectation of Bang, Whop, go gut him out. That cost me a fairly nice buck, one season. I had pulled a tad low, and the Whop was a high hit on the near front leg bone. As I was walking up to gut my "dead" deer, he jumped up and boogied. I unslung my rifle and cut down on him and got an eyeful of 4X setting sun. Bummer.
I lost one other buck that I'd hit with a chest hit. Outside of that, the other forty or so were pretty much one-shot DRT. A few, yeah, coup de grace, but they were just dead and didn't yet know it.
Most folks I've hunted with were pretty much the same way as to expectations...
Art