If we are going to have one DH....why not have nine?
The same reason we don't say, "If you're going to allow one pinch runner... why not allow infinite pinch runners?": roster sizes.
A position player plays the vast majority of his team's games; a pitcher does not. And due to the way DH rules work, allowing a pitcher to bat in games they do and do not pitch in hurts the team (without these particular rules, there is significant potential for gamesmanship shenanigans). It's ultimately easier to allow a team to substitute a DH for a player that won't even play back-to-back games.