This debate, conversation, whatever, could go on forever.
I think there are plenty of good ideas in this thread depending on your abilities as a reloader or not.
Personally, as a guy who doesn't reload and doesn't want to spend a ton of money on premium grade bullets, I am particular to just a few calibers based on versatility, value, and ease of locating ammo.
First and foremost, if you're not worried about a bit of coat damage on coyotes (or shoulder fatigue from a busy day), the 30-06 is agreeably one of the most versatile rounds out there, with plenty of "over the counter" ammo choices.
Second would be the .25-06, which is flat-shooting rifle that is very handy on deer, coyotes, etc. Ammo may not be as readily available as 30-06, but isn't very hard to find either.
Third would be the .243. It's a fine rifle for deer and varmints to reasonable ranges. Also very common over the counter. Prob as common as the 30-06 in most retail stores.
I would agree that I've heard plenty enough negativity about the .243 Handi that I'd shy away from it myself. Personally, (and yes I know it's not a Handi) the best and most beautiful rifle I've ever owned was a Winchester model 70 Lightweight in .243. That was a fine, extremely accurate rifle! It was light enough as well that youngsters could handle it. I still kick myself for selling that gun, and it's been nearly 15 years!