Anyone here hunt deer with those more expensive, larger caliber air guns?

I think it's mostly a stunt. If you look at the velocities the big bores produce, it's like hunting with a .38 Special or maybe a low power .45 Colt load in a revolver. Will it kill stuff? Sure. Is it a practical and ethical hunting tool for large game? Not so much by my standards.

I don't believe it is legal around here. I wouldn't mind using an air rifle for coyotes, given the lack of restrictions on moderating sound on an air rifle I think it would be great to shoot a .45 colt level rifle with no hearing protection without completely breaking the bank and jumping through hoops.

Check out the Fieldsports line of British videos on You Tube. Lots of airgun stuff.
Watched a test of a .30 airgun on fox heads. Pretty scary.

Air rifles are legal for deer in Missouri, and work if you can get close enough. A .44 carbine has twice the range and half the weight, but an air rifle (AR?) would be fun if you want a challenge.

SHOTGUN NEWS has an excellent review of the new AirForce TEXAN... .45 caliber and will run bullets up to 1000 fps... I have a Condor SS right now and will be saving my pennies for one of these...

Bob