Here's a fun idea.
According to Jayson Stark of The Athletic, MLB decision makers have at least discussed a potentially interesting solution to making the DH universal:
"But wait! Here’s one possible wrinkle that has been kicked around in behind-the-scene brainstorming sessions: How about a rule that would allow teams to use their DH only as long as their starting pitcher remained in the game? Then, once the starting pitcher exited, that game would revert to old-school rules."
So, imagine this scenario: Max Scherzer is starting for the Washington Nationals in 2020, and rather than him batting, a DH would hit for him as long as he would be in the game. While this solution wouldn't necessarily increase pace of play, it would increase offensive output and prevent pitchers from getting injured in the batter's box or running the bases.
However, the strategy of a manager having to decide how long to keep a starting pitcher in the game based who bats out of the pitcher's spot in the lineup wouldn't go away. In fact, it would probably become more interesting.
If you want to take your starting pitcher out after five innings, that's fine. It could also mean taking a bat like Edgar Martinez, David Ortiz or Nelson Cruz, probably the best hitter on your team, out of your lineup. Otherwise, you would have to find a spot for them to play defensively, and a DH is typically a DH because they are a liability in the field, especially in late-game situations.
What would also be interesting is the effect this would have on teams that use the opener. By using the opener - who, at most, stays in the game for one turn through the lineup - teams would essentially be forfeiting their chance to employ a single DH, rather than pitchers or a combination of pinch hitters.
It's unclear, if implemented, whether this solution would apply to both leagues, or just be a way to implement baseball into the National League. Frankly, most American League fans are content with the current DH arrangement. The DH has existed in American League baseball since 1973, so most fans of American League teams have no recollection of pitchers hitting for their teams. Many National League fans, though, enjoy the strategy of the manager having to weigh when to remove his starting pitcher based on how much of an offensive edge could be gained by pinch hitting.
Stark acknowledged in his piece that such a solution may be a long shot to become an actual reality. There has, however, been credible speculation that baseball will consider making the DH universal after the current CBA expires following the 2021 season.