Vastly Superior? How do you measure Vastly Superior?
Since the introduction of the DH in 1973 the AL has won 21 WS and the NL has won 16. Five of those 21 were the Yanks and they are in a league of their own as they have dominated the WS before and after the DH. Take them out and both have won 16 since the DH
Prior to the DH the AL won 41 WS (22 by the Yanks alone) and the NL won 28. Take the Yanks out and it is 19 to 28. Not much has changed since the DH introduction so the DH has had little affect on changing the superiority of one league over the other when it comes to WS championships.
Dominance by attendance? Last year the NL averaged 31,437 people per game (home and away) and the AL averaged 28,863 per game (home and away). The snoozer fest is attracting more people to their games. Take the Yanks out (the only team to average over 40,000 per game) and the difference is even greater. The bottom 5 teams were all AL teams. 6 of the 10 top attendance teams were NL teams.
Runs and pitching comparison. Last year the NL scored more runs on average than the AL, but since the DH and before the AL has on the average scored slightly more, meaning the DH has had little affect with the AL average about 0.2 more runs per game. 2 tenths of a run is not very noticeable, let alone dominating. Replacing the worst hitter on the team with the best hitter in many cases only gives a team about 1/4 of a run more per game.