There's some dubious logic here IMO. For 150 games AL teams can throw most of their money into 9 position players and can largely skimp on their bench players with the lone exception being C where most teams will run two decent catchers. For example, let's look at the Blue Jay's roster. Their starting 9 is Martin, Smoak, Devon Travis, Tulow, Donaldson, Saunders, Pillar, Bautista and Encarnacion. Their next 4 players are Josh Thole their back up catcher, Darwin Barney(making $1.05 mil), Ezequiel Carrera(pre-arb), and Ryan Goins(league min). Cleveland like wise features a bench of Rajai Davis($5.25 mil), Marlon Byrd($1 mil), Chris Gimenez(back up C $1 milish) and Abraham Almonte(pre arb). In other words, AL teams don't need as deep of a bench because they rarely if ever pinch hit. In contrast, NL teams will often employ more expensive players on their bench because those players need to be able to field semi-regularly.
Also, you're missing one aspect of the DH which is greatly helpful to AL teams. They can take fading stars and plug them into a DH position when they no longer play like they are being paid. For example, Victor Martinez is still a useful hitter but he couldn't play anywhere in the field at this point. Like wise, ARod was getting huge money the past few years but they didn't want him playing the field. It's rarely a case of the team going out and paying the best hitter in the league to DH. It's more often the case of them having paid some position player big money in the past and now having a get out of jail free card. NL teams in the same position are forced into playing that aging star in the field
