I have used a lot of different varmint rifles for coyotes, and currently use .243 Win., 6.5 Grendel, .223, .20 Practical, 6br, .22 Nosler and .257 Roberts. They all have merit. Within 400 yards, most anything will work in center fire, though I usually gravitate towards the .243 and 6BR when shots are expected to be longer - and .223 within 275 yards or so and .22 Nosler to 400 or so. Not that they can't hit further - just that they seem to still retain lots of killing speed at those distances. I run 58 V-Maxes in the .243 at 3,300 FPS for best accuracy from that particular rifle.
I think I like the .22 Nosler for the best "all around" gun, in that it is somewhat of a bridge between the .22/250 or .243 and the .223. I don't mind giving up a bit of velocity to the larger cartridges when I get reduced muzzle blast, longer barrel wear, 20% less powder used, etc. I'd use my 6BR more if it were not so heavy. The .22 Nosler brass is only available from Nosler and under the name of Dogtown, whereas the 6BR brass is somewhat limited in who makes it as well - and most of it is a bit on the expensive side. Buy enough of whatever you get to run your barrel down - then it is not a concern as to availability. I run 55 Blitz Kings at 3,365 FPS from the .22 Nosler, as well as 62 ELDVT's at 3,030 FPS. Inside 400 yards, the 55 Blitz Kings beat the 62 ELD on bullet drop. Note the 62 is moving only about 10% slower than my .243, while shooting a heavier bullet - and recoils no more than a .223.
If I were to have only one coyote gun - it would likely be the .243, or a close variant like the 6 Creedmoor, being able to do double duty on small deer, pigs, etc. I can load it with 58 V-Max for super flat shooting or load up on heavier 100 grain bullets when needed. There really are no clear winners in this varmint game. Your Grendel would be well enough, for sure, with a light bullet to get that velocity up a bit.