[Starfield]

How's this for an alternative to the DH?

Simply have the 9th spot in the rotation be a non-batting spot. The batting order goes from 8th to 1st. Any player can be put in that spot, but naturally it will be the pitcher 99.99% of the time. What are the downsides?

It would affect stats since each player will have more at bats a season.

It would tighten up the batting order a little bit. You only need 8 hitters to fill in the lineup. This is in contrast to NL lineups that are a little weaker for having the pitcher hit and AL lineups that are a little stronger for having a dedicated batting specialist.