There's honestly no reason for Ash to not DO the gym badge quest.
People might think its overdone, or as Champion Ash has better things to do.
Couple of things:
1. It's still battling, and would be a great way for Ash to train his Pokemon at the very least.
2. Tobias. Tobias steamrolled through the gyms with JUST Darkrai (perhaps even more impressive than Alain's Mega Charizard X, hard to tell though). If Tobias was perhaps on Kukui's level or end of Ash Alolan champion level, or potentially even higher than that (which I believe he would've had to be), it didn't stop him from going through the gyms and the Pokemon league in the Sinnoh region.
There would be literally NO reason not to have Ash do gyms, unless Go decides to do them, and Ash is entering something a lot more difficult than merely battling gym leaders and collecting badges.
3. Perhaps even Leon himself wants Ash to go through the gyms in Galar. Could Ash really deny the world champion in a way that would make any sense if it leads to Ash rematching Leon?
Yes and there seems to be a pattern that they get to spend two episodes in Galar so Ash can easily finish a gym battle in two episodes with one episode that focuses on Ash meeting the gym leader and then the episode after that the actual battle. Plus because they are constantly going to different regions they easily just show up in the city that has a gym.
This especially, as they are region hopping, they don't need to spend a majority of the series in Galar for Ash's gyms, he could easily go where Sakuragi wants them to be, and when Sakuragi doesn't have anything more for Ash to do or even Go, they can go to the next place where Ash gets a badge and then Sakuragi needs them to go to Kalos for example.