I think it has been quite some time since teams have routinely given all of their DH at bats to overpaid superannuated sluggers.
Here is a quick table I put together from Fangraphs data for 2024, showing how many total DH PAs each team had, how many they gave to their top DH, and at percent of the teams total that made up:
Only 13 of the 30 teams gave more than half of their DH plate appearances to one guy, and 12 other teams gave fewer than a third to their top DH.
Here's another way of looking at it: a total of 361 different hitters DH'ed for their team. Here is a list of all the players who "half-qualified" for a batting title (i.e., minimum 251 plate appearances) and got more than 50% of their team's DH plate trips:
You don't have to count, I'll just tell you: only sixteen of the half-qualified hitters who DH'ed at any point of the season received as many as half the DH at bats their team logged last season.
We can conclude that while there were some old, slow-footed, overpaid sluggers on these lists (hello, J.D., my old friend!), most of these guys were still under 30.