My testing showed that wads can make a substantial difference in pattern density (total spread) at various distances, and some difference in uniformity. I found differences between some brands and types of conventional wads, not to mention wads like Flite Control or Ballistic Products. An extreme case, my HD gun with the Full choke tube will shoot S&B factory 00 buckshot loads insanely wide -- like 15-18" at 12-15 yards! -- vs. a fist-size group with Flite Control wads. (S&B buckshot has the most minimal "wad" of anything I've seen in decades.)
I agree. The reason I mentioned muzzle velocity (though I didn't elaborate) was just as you said. Based upon my testing, lower muzzle velocity is my preferred in almost every case. The pattern holds together enough better that you can clearly see it on the pattern board (most likely a function of aerodynamics), AND by the time you get out to 40 or 50 yards where people naturally THINK the extra MV would help, the shot charge has lost a big portion of the extra MV that it started with, AND the pattern is worse. I'd rather hit the target with a more uniform pattern traveling a bit slower than to have holes opened up in the pattern. Again, you can see the result on the pattern board. And the lower recoil makes for faster subsequent shots.
Agreed on the equal payload point, and that's why I own and often use the 20 gauge -- because it's perfectly adequate and even preferable for some shotgun tasks. It has the advantages of lighter weight, lower recoil, and lower cost to reload. But ultimately, if range is of significant importance, the additional pellet capacity of a 12 gauge can provide for greater effective reach than the smaller gauges, all else being equal.