How "Hard" a shotgun shoots is pretty well defined by two factors, Penetration of the shot & sufficient pattern density to insure a hit. Otherwise a .22 Short would shoot "Harder than most any shotgun ever made But you gotta hit it.
Heavier shot loads, assuming similar choking, provide denser patterns increasing the odds of a vital spot being struck. "No One", no matter how good, can aim an individual pellet from a shot load.
"Hard" is not truly the best word to use, but to use it as commonly meant, Will a 1¼ oz loaded shoot "Harder" than a ¼ oz .410 load all other things being equal, "YES", will a 3 oz load from a 8 gauge shoot "Harder" than that 1¼ oz 12, "YES"

Miller/TN
I Didn't Say Everything I Said, Yogi Berra