I wouldn't consider myself anywhere near an expert on the subject of gator hunting, so take this with a huge grain of salt, but having seen every episode of Swamp People to date have two comments to the above.
1) Near as I can tell, almost every one of the hunters on the show is using 22mag not lr. That's of course not to say that it isn't or can't be done with 22lr.
2) i think they mostly use rifles to keep their arms as far away from the gator's mouth as possible when lining up the shot. Using an 18"-22" barrel, you can basically stick the end of the barrel right above the "kill spot" on the back of the skull. I'd think most people would be very hesitant to attempt this with a handgun unless absolutely necessary. A few have used a handgun on occasion when needed including either a Taurus or S&W revolver (couldn't tell which with the limited size and angle of the camera), and another guy used an NAA mini once when he fell in the water.
Also, some of the guys (and gals) on the show seem to be very good shots and using a rifle they can take a surfaced alligator from a decent range. Not sure how far they could do this with a handgun.
JMHO from watching the show, never even seen a gator in the wild yet (though I hope I can some day). So again, take my analysis with a grain of salt and feel free to correct me if you any of you know something I don't.