You really need a choked barrel for bird or running game with shot to get effective range.
I would say it depends on where, what, and how you hunt. I used to hunt with a Winchester 1400 with the Poly Choke cut off, and my uncle has a NEF smoothbore 10 gauge slug gun with no sights. Based on these two guns used during a year's worth of days, I'd say that a choked barrel isn't as necessary as most would think.
The following are my own observations with bird hunting, but since my hunting experiences, methods, and habitat might not jibe with yours, what I would use for pheasants might not jibe with your opinions...
I would say 75% of the prairie chicken and sharptail hunting I have done would be better accomplished with no choke compared to a modified. Sunny days and no wind, and they get up within ten yards. Cloudy days and wind and you'll be seeing 40 and 50 yard flushes, lots of luck with any choke.
Ducks coming to the decoys, most are under 25 yards, and most nontoxic shot patterns tighter than lead, so you would be okay with no choke. Jump shooting, depends on how sneaky you are. I wouldn't feel handicapped without a choke.
Geese, probably need a choked barrel, at least with the luck I have decoying them...
Quail with a dog, or even without one, one doesn't need a choke.
Pheasants with a pointer, don't need one, pheasants without a dog, I'd say yes. Late season pheasants with or without a dog, I'd take a modified.
Doves definitely need a choke...
Hungarian Partridge, not a lot of experience with the few around here, but they get up far enough away to need a choke.
Turkey is supposedly a no brainer, but the big guy in the picture got his two birds one weekend with his NEF single shot 10 gauge slug gun (no choke) and Federal Premium #4s:
He doesn't shoot if they aren't within 20 yards of his decoy.
Some shotshell loads seem to have "choke" built into the wads and buffering. Federal Flightcontrol wads are an example. One summer years ago, I patterned shotguns and loads until my shoulder started going numb from the turkey loads. 90% of what I saw would be what anyone would expect, but then some patterns defied logic. One of my brother's Ithaca M37s would always put more Federal Premium #2s into a 6" circle at 30 yards than Federal #4s, time and again. Shot up a whole box of each, because I knew that the #4 load had a lot more pellets.
If all I had was a 12 gauge with no choke, it wouldn't bother me a bit to hunt any gamebird, but I would sure run a ton of shells through my gun and see if I could find some loads that would tighten up patterns when needed.