Listen. If I want to kill coyotes and fox because I'm trying to sell you something, have a TV show or protecting my herd and I didn't care about fur. My first choice for a weapon is not a 17HH or any other 17 cal. It would be an AR in a 20 cal. with Thermal and I could shoot more then 1 or 2 coyotes coming in, and if only 1 came in and I missed with the first shot I could unload the clip. Now if I wanted to play sniper and lived in open country where shots could be taken as far I could see, I still wouldn't want a 17 cal. in anything. I would probably get a 6.5 Creedmoor and sight in for 1200yds.
Now back in 12/31/18 Artic Wolf put a video on Fox pro where he put a 243 87gr Vmax thru a coyotes shoulder and it excited thru the bottom paunch. That coyote traveled with guts coming out for 250yds. I myself have shot coyotes with 243 85gr.BTHP and blew holes big as your head thru them and they never twitched.
Every caliber that has been shot and will be shot, will kill coyotes if its a vital shot. Every cal. will loose animals so long as the shooter doesn't use the right bullet, within its proper range, with proper shot placement.
In regards to Cal. and bullet that will do the least damage to fur and Drop them where they stand. Its not so simple.
If the rounds are traveling over 3000fps you will tear fur sooner or later. If rounds are less then 3000fps you can avoid fur damage usually, but you have to be more precise with shot placement and range is limited.
Having the right bullet and putting it where it needs to go to Drop them where they stand requires more then just buying a 22-250 and shooting a 50gr.Vmax . Same holds true with smaller Cals. like 22 Mag. 17HMR, 17WSM, 17HH etc..
I have watched old black and white films that Gordon Eastman took of Eskimos shooting Polar bears with 22mag. They would get sneak up on the bear and shoot them in the temple. DRT.
So don't pay no mind to all the people that say such and such is not a coyote Cal. Its because of them or their buddy not using the proper bullet, within its effect range and placing it where it has to go.