What is your limit? A used NEF single shot $150 will do what you want. A used Winchester 1300 $250 will work. An old 870 Wingmaster $ 250 will work. Nothing wrong with Browning, Ithica or Bennelli. Note the new shotguns from Turkey from Hatsan, Winchester and Weatherby also work well. A $1000 will get you a used over and under Browning that will last you a life time.
You ask for a rifled barrel for bird? weird. Rifled barreled shotguns are for slugs not shot. A 20" barreled IC choke shotgun is good for 100 yards with slug. 200 yards with accuracy with rifled barrel. I've shot deer with a single shot shotgun with a 28" barrel and full choke. Works, but don't expect it to be fun and comfortable. I like many others have multiple shotguns with preferences for certain tasks.
Field birds I prefer a side x side 20ga. Nice and light, swings fast.
Deer 12 ga pump rifled barrel with scope
Duck 12ga pump
Piss around take camping in the boat a Marine finish 12 ga. (Picked up a Hatsan myself)
It all comes down to you and what you like. I hate o-rings in semis
If you decide to get into action shooting a Benelli M2, or M4. If just for fun and giggles, a Hatsan MPA TS.
So you can pick up a shotgun used for $150 or spend over $4000 new, it's up to you. You wouldn't use a 32oz framing hammer to hit finishing nails. Same goes for all the different variants of shotguns.
What is your limit to spend?
Criteria;
Reliable
Available and reasonably priced rifled barrel
Use:
Killing birds,,,,WON'T WORK
clays, ... WON'T WORK
and perhaps some deer Will WORK