You'd think this would be easy to figure out.

[Bittereinder]

With a .22? Sort of normal not to be able to see the actual hole. You should be able to see a mark on the target though. Still best t get some kind of optic. Binoculars(likely the least expensive option. Cheap compact binos start at $21.99 at Cabela's. $70ish for better quality. Bushnell 10x 42 binoculars are currently on sale at Crappy Tire. $100 binos for $69.97.), spotting scope, etc. Even an old scope sight will do. Assuming you have such a thing.
"...your grip might change from shot to shot..." Not with a properly set up spotting scope. Said scope gets mounted on your shooting box so you just move your head. More expensive though.

I'm talking about grip on the gun, not the scope. I'm just recommending against putting down the gun after each shot, which I think would make it hard for anyone, especially someone new to shooting, so understand how large their groups are and where they're hitting, to help them adjust sights and identify potential areas to improve technique. Perhaps bulls-eye shooters check after each shot, I don't know, but that's different.