That 1 1/4 oz of #5 shot load you tried is really a 12 ga. load. You might want to back it down to what V>M. Starr said, and see how good a pattern you get.
Speed does not contribute to good patterns in an open cylinder, or cylinder bore gun. Birds are killed by the amount of energy delivered by more than one pellet to them when they are shot out of the air. The tighter the pattern, the greater the likelihood that you will hit the bird with multiple pellets. Like Round Ball, shotgun pellet are slowed dramatically in the air, and it is the size of the shot, PLUS the corresponding weight that delivers energy down range. If you check the tables in the Lyman Reloading manual for shotguns, you will see that regardless of how fast the shot leaves the muzzle, all the velocity is pretty will lost in the first 20 yds. By the time you get to 40 yds, the remaining velocity difference between a " hot " load, and a " mild " or " slow " load, are insignificant, and are within the variations you can get from shot to shot. Velocity creates unwanted recoil- no test of manhood, really- and blows the patterns, making misses much more likely. The loads V.M. Starr recommends are almost all below the speed of sound( 1100fps) which is pretty slow by modern ammo standards. But, they kill birds.