That means a bunch of the world will spend most of their “daytime” in traditionally nighttime hours
No, no one would do this. You would continue living your life when the sun is up, the number on the clock would just be different.
Islam requires regular prayer in the direction of mecca and plenty of nations have Islamic law.
So just continue doing this based on the previous schedule? Many religions still celebrate holidays based on alternate calendars and many holidays have strange rules for when they occur. This seems like an incredibly minor issue to me?
We changed how clocks work
Yeah, I agree that the change would be so painful that it isn’t worth it. I am just arguing that I think the end result would be better. Not much better, but better.
you can pretty reasonably assume when people will be around and doing stuff by looking at their time
This seems like a very artificial problem. When will you know their time previously but not their location or relative time of day. You will still know what people are doing. Just because you add the magic number based on their location in the world before consulting their schedule instead of after doesn’t change anything. This only seems like a problem if you were magically teleported to another location underground and only have access to a clock.