From sometime in the mid 1970s until perhaps 1981 or so, my only repeating shotgun was a Remington model 17. "All Around" use would be limited in scope in my world at the time, but, included early season ducks (lead was legal) grouse, woodcock, pheasants, and a very occasional bunny. I had a friend who had a Basset hound (!) that we would use to run rabbits-we were young (didn't know any better) it worked better than you might imagine, and I learned hounds are for other people besides me. My pop frowned on folks who used anything but a .22 for squirrels, and I shot bunchs of them with a few different rimfire rifles, but, that doesn't count. Everyone always said that 20 was a bit light for roosters, but, nobody ever told the roosters that, I got more than my fair share. I did buy an 1100 12 in the early 80s, and got sick of lugging the damn thing around. Ditto a Red Label 20.

My definition of all round hasn't changed much (I don't shoot bunnies at all now that I have bird dogs, and ducks are a fading memory) and from where I am sitting, I think a pump in 20 or 16 with a barrel or two makes a lot of sense in the above version of all round, with the qualifier of $300 or less attached. Holding that thought, as much as I love the 16, buying a 20 avoids some easter egg hunts for ammunition, and would thus be simpler/cheaper to feed. I don't hunt ducks, but, have heard that TM shot has made the 20 a contender again in a duck slew. Carrying a 12 pumper or auto all day in hilly or mountainous terrain was never my cup of tea. SE MN, around Brownsville or Hokah, taught me that.

I don't own just one gun, but, there are days I wish I did. I think an Ithaca 37 in 20 or 16, could pull it off. At least for me.
I have more guns than time to use them, and there are days I think of peddling all but two or three and devoting time to becomming better at what I shoot with just a few. I'm having one of those days today.
I didn't put a $300 dollar limit on it, but my All Round gun would be either of the Darnes, a 12 or a 20, that live in the safe. Useful, different, not sensitive to ammunition, light, handy, graceful. Any double you can shoot well would certainly qualify, however.
Best,
Ted