I've heard that it's more about round count than anything else, but that's just what I've heard. I don't own a double, and probably won't if 870 reliability in a 2-shooter can't be had under $3000. It'd be nice to see a serviceable gun under 500. Just remember, you don't have to have a double for defense.
Joe Biden was wrong about a lot!
I'm happy with my 870 for the do-anything gun. Get a 20" rem-choke deer barrel with rifle sights, and you can change chokes and loads to kill everything from dove and squirrel to elk and moose. Mine will print acceptable groups with the I/C choke with slugs so I can be prepared for anything when woods bumming. That same gun also stays loaded with 00 buck for HD. I've priced wingmasters in good used condition as I'm looking for a 20 ga. The used 870s are within your budget. A barrel will add about $150-$200 to that price, though. My buddy's brother-in-law swears by the 500/590 series and uses a 500 "tactical" for woods bummin. So far, the impact on the squirrel population is noticeable! Not sure about battlefield tactics.