If I had to take one gun with one barrel with me and have to survive I'd take my Mossberg 500 with a fixed Modified choke. The 28 inch barrel is a bit long for heavy bush, but i have taken rabbits, ducks, phesant, and i wouldn't be shy to take a deer or defed myself from wild animals with the right ammunition.
Of course OP is not talking about this exact situation, so multiple barrels and chokes make a shotgun more versatile, they can make it more complicated. With my 500 I never second guess what barrel or choke I have available when a shot presents itself.
Out after bunnies and feel threatened? shovel some buckshot in there.
Sitting under ann apple tree trying to get that phesant that always gets away, but a deer staggers out into the choppings? I got an answer for that too.
This assumes that hunting seasons overlap where you are. I'm sure the DNR could turn a blind eye if you were lost for a week and bagged a deer but don't quote me on that.
My deciding factor for 500 over 870 is simply the ambidextrous nature of it. I'm left handed and I find them a little easier to operate. Both fantastic affordable platforms that will not let you down.