One gun to rule them all, to do all things for you ... this is a long-asked question.
You have to accept that you will have to be compromising on something, somewhere.
First - yes, you can kill a coyote with a .22 LR. You can kill just about anything with a .22 LR. That does not mean that it is the optimal choice, however, and it will definitely limit your range.
What do you want to do with the rifle the most? If you want to hunt deer most, then you have to focus on capable deer rifle calibers. The .30-30 is a very capable deer rifle, but it does have limited range - 200 yards or so, for most people. If that is far enough for where you hunt, then it is a fine choice, and will easily take BLACK bears, too. Might not be enough for Brown or Grizzly or Polar bears, but then again, they have been killed with some oddly light calibers. ETA: Coyotes aren't a problem for the .30-30, either, within range.
If you want to hunt bears the most, then your rifle might kick a bit more than the .30-30, because you'll probably want something that hits a bit harder, like a .35 Rem, .444, or .45-70.
.30'06 is a fine cartridge, but it annoys me to shoot, too.
If you want it to be a capable deer AND coyote/groundhog rifle, then you are going to have to focus on another type of cartridge: the dual-purpose cartridge. They've been around for a long time, starting about with the .250 Savage, also called the .250-3000, because it was the first cartridge that would throw bullets 3000 fps. The 6mm/.243 through 6.5mm/.260 class is what I'm talking about. There are a lot of cartridges in this group, because there are a lot of people in the world who think like you do, that there should be a rifle cartridge that will do in a deer and then turn around and whack varmints and fur-bearers, no need for two rifles. Most of those cartridges, loaded right, should do in black bears, too.
So, in that class, you have the .243 Win, 6mm Rem, .250 Savage, .257 Roberts, .25-06, .260 Rem, 6.5 x 55mm Swede, and others. There are also Weatherby cartridges in the same diameters, but most people find them too expensive to feed, and the ones mentioned are standard cartridges that you can find ammo for in more places that will do whatever needs doing anyway.
If you go with a heavier caliber, like the .308, you will definitely be killing your deer and bears well (and anything else in North America), but it may be a bit heavy for coyotes and groundhogs and such, and such cartridges definitely kick harder than the dual-role cartridges I laid out above. Also, if you want to be able to keep hides, then you want a lighter caliber for coyotes, bobcats, and so forth.
Basically, you have to set your priorities, and use those priorities as guidelines for picking out your caliber.
My advice? If, as you stated, you want to do coyotes and deer as your primary game, and be able to buy off-the-shelf ammo just about anywhere, get a .243 or a .25-06 and practice shooting. Myself, in this class, I have a .243, and it's a great round.