[TheSollyLama]

wow, that's a strange sort of set of posts.

You don't need 'shock' to kill a coyote. But a frangible bullet moving at high velocity that is unlikely to leave an exit wound is better.

So the most usual candidates are .223 and .22-250, with more rare examples being .222 Swift or .221 Fireball. The newer .204 Ruger is supposed to be great too and even easier on barrels and virtually zero recoil.
Super velocity with a ballistic tip bullet that all but explodes on impact.

A .22 rimfire is marginal in the best of circumstances. A .22 Magnum is a proven yote killer, but usually only out to 100 or so yards for a clean kill.

The advice is centerfire .22 for coyote. I personally use a Howa 1500 I bought as a cheap combo knowing I'd do work to it over time.

I chose .22-250 for a caliber, but .223 can be a couple dollars a box less. Don't reload so I use Remington Accutip factory rounds.

With coyote people usually hunt for either fur or because they are ranchers. If you want fur, then the .22 caliber centerfires (or the .204) are probably your best bet for finding a gun and finding ammo.

Ranchers could care less about pelts and will shoot coyotes with whatever is handy. Any 'deer' gun will blow a coyote into a fine red mist without investing in a new gun.