The .22-250 is a tried and true varmint round that has been around for a long time. I have had several and loved every single one of them. I have an old Remington 700 BDL with a 12 twist that i shot the 36 grain Barnes VG out of that was screaming well over 4000 fps. That combo took a lot of prarie dogs and other small varmints. I also never hesitated launching them at coyotes. It is a cool bullet that takes the ricochet concern into account. That rifle also shot the 55 grain bullets very well.
I currently shoot a 22 BR with a 7 twist barrel that I have a load for Berger 80 VLD's and 55 grain Berger HPFB. Both bullets shoot 1 hole groups on a regular basis. The 22 BR is a little slower than the 22-250, but is a riot to shoot. If you want to try something new I would highly recommend it!!
If you want a 204, then get a 204. It is pretty much that simple. The 22 cals are probably a better "all-around" choice. But I personally wouldn't worry about throwing lead at a coyote with a 204.
When deciding on a new rifle, I think we tend to over complicate things. But we do that because we can, and its a lot of fun. Either route you are thinking of will work great for your application. I say that if the 204 is calling to you, go get one and have fun with it. If you are concerned with it not suiting your needs, stop worrying and shoot