Hhmm - - interesting thought.
A .22LR is good on everything up to coyote in size, depending on the distance. It's also fairly quiet on its own so it is less likely to disturb other game animals (or the neighbors).
A .243 is a good, accurate rifle that can be handled by young and old alike. It can also overlap the .22LR and go down to groundhogs or up to a small elk (again, at closer range). Granted, a .308 or 30-06 would be better for the larger herbivores (elk/moose), I question if the .308 is enough gun for dangerous game except at dangerously close ranges.
A 12 gauge shotgun can overlap both the .22LR and the .308 when you consider its range limitations. It can take the same small game and varmits as the .22LR with "lo-brass" loads at close range (quail, squirrels), larger game (groundhog, coyote, pheasant, geese) at longer ranges with "hi-brass" loads, and, with a rifled barrel and sabot slugs, large or even dangerous game, but the latter at uncomfortably close range.
All in all, the .22LR, 12 ga., and .243/.308/30-06 will take care of everything in the Lower 48.