With the universal DH likely in the 2022 season, note that the trend of pitchers batting in NL games shows a slow and steady decline over the last 48 years. This is based on event data since 1974, counting plate appearances where the defensive position of the player is pitcher. It does not count pinch hit appearances by players with the primary position of pitcher. The year you would see pitchers come to the plate the most was 1975, 5.6 PA per game (both teams). The low came in the 2021 season 3.9 PA per game. (The graph in the spreadsheet excludes the 2020 season, clearly an outlier.)

Pitcher PA per game is declining at a slow rate of -0.026 per season. I suspect, however, that the opener strategy and limiting starters to twice through the batting order continue to take hold, we would see an even faster decline without a universal DH. We would get very close to the “pinch hit for the pitcher every time” strategy.