How about a compromise?
You can pinch hit for the pitcher whenever you want, but the batter who hits for him has to leave the game or replace someone on the field. Pitcher doesn’t get removed. Want to hit for the P the next time he is up, same rule, the batter leaves the game or someone else on the field leaves the game. So no more P batting with two on and two out in a one run game and you don’t have to pull him, either.
I think the prior arguments for making the DH universal across both leagues are pretty compelling. With the increased role specialization of the game, it's not fair to expect pitchers to swing the bat all that well. Having a DH to hit in the pitcher's spot makes for more competitive at bats. Consistent rules across both leagues should also negate any relative roster-building advantages or disadvantages, especially come playoff time.
That being said, I like the idea of instituting a compromise solution that preserves some of the NL strategic trade-off that comes from a pitching change. My suggestion would be to staple a specific DH to the pitcher of record completing an inning. So, if the starting pitcher gets pulled in the middle of an inning, so too does the DH. Any new reliever closing out an inning requires a new DH. NL managerial purists could still opt to have pitchers bat (a pitcher could serve as his own DH). Of course, the DH doesn't have to be designated until the half inning after the pitcher makes an appearance (so an inning with multiple pitching changes only requires a single DH change). Also, as before, any team having to use their DH in the field automatically loses the DH (a.k.a., from that point forward, that team's pitchers have to serve as their own DH's).
Consider some of the implications:
1. In a game where the starter is dominant, teams get the added benefit of using their best DH deep into the game. An example of this would be stapling Big Papi as DH to 2007 post-season Josh Beckett.
2. In a game where the starter gets in trouble early, teams have the added disadvantage of losing their best DH early in the game. Think of 2018 ALDS Game 2 or 2018 ALCS Game 1 where short outings by Price and Sale lead to games where JDM only gets 1-2 PA's.
3. In a Tampa or Milwaukee game where a reliever is used as an opener, the Rays/Brewers would likely want to save their best DH for after the pitching change. This might mean that they would also potentially need an "opener" DH (unless they plan for their starter to only go an inning, and they bat the DH close to the bottom of the batting order).
4. In a postseason game where Joe Maddon wants to substitute pitchers like crazy, he will likely exhaust his entire bench of positional players (just as he did in this year's wild card game against the Rockies).