Where's Troy Landry when you need him? Choot 'em!
The stories of Mr. Landry killing gators with a .22 are almost certainly true. There also many stories of elephants being killed with .22s.
But here's the deal: The people doing that know the animal's anatomy perfectly and are taking slow, controlled shots. They are
not taking snap shots in a high stress environment.
Alligators, like all reptilians, have tiny brains that are hard to hit. That's why, earlier in this thread, I suggested shotguns (with buckshot) or high-powered rifles to take gators. The high-powered rifles will almost always inflict hydrostatic shock, which will put the animal's lights out in the case of near miss cranial shots. Buckshot patterns are not going to expand much at close ranges, but they do open up somewhat more upon contact with bone and flesh and create traumatic "smashing" wounds.
Ms. Kathy's suggestion of using slugs isn't bad, but it has two drawbacks: Slug velocities (unlike high-powered rifles) aren't usually high enough to cause significant hydrostatic shock damage and slugs require better accuracy than buckshot.
On many occasions I've taken my 16' canoe down small bayous and canals which had gators sunning themselves on the banks. I have a little old 1.5 HP Johnson outboard that I use with the canoe, so my approach isn't silent. The gators often ignore me or slide into the water and sound upon my approach. I'd be lying if I said this didn't bother me, but nothing untoward ever happened. I've spoken to a lot of old timers around the marshes and no one ever heard of a gator attacking a boat. This isn't to say it's impossible, but the chances seem to be vanishingly small.
My wisdom regarding gators can be summed up very succinctly by saying that you shouldn't keep them as pets, but if you do...don't let them sleep in the bed with you!
Best
Doc