No, that shouldn't be the end result.
NL teams face a greater disadvantage by playing against AL teams with a DH. AL teams carry a DH full time. NL teams end up using bench players who usually aren't up to par offensively with the AL DH on the other side.
The rule needs to be standardized full time. As a traditionalist I'd prefer to see the DH go away. As a realist, given the power of the MLBPA and the fact that every other level of baseball allows the DH, I suspect that the DH will soon be implemented league-wide.
You are correct that the DH gives the AL a distinct advantage over the NL. Not only do AL teams carry one more starting caliber hitter on their rosters, the presence of the DH also allows them to sign aging players and have the option of moving them to DH as they age. NL teams don't have that option, so they tend to not sign the aging sluggers that are a liability on defense.
This advantage slowly brings more talent to the AL, and has been bared out by the AL's dominance in interleague play.