What you need are good laws, not so much a 4 day work week. I just go to the doctor during office hours and tell my employer I have to go. I even get paid time off for it, like everyone else working in this country. Same for the dentist or any other kind of medical thing.
Sure, it’s not always optimal if you work in some sort of shift, but they are required to make sure you can go.
By the way, not that I wouldn’t appreciate a 4 day work week, but this seems like a bit of a stretch to say that this is the reason why you would need one.
I run a similar setup, but with syncthing as the syncing system. Every time I connect the phone to the charger it just syncs the database and I can even sync it outside the home network. Works like a charm. Worst case you get a sync conflict which is easy to solve.