Re: Just set the entire moon to Coordinated Universal Time (UTC +0) ...
Unfortunately, even over the cosmologically short distances from Earth to Moon, relativistic effects can be observed. Even much-closer by GPS satellites have correct for relativistic effects to generate decently accurate position information.
So, saying sync to UTC is missing a whole bunch of thinking needed.
An agreed terminator on the moon getting it's own time reference solves a bunch of problems that most people won't ever have.
Setting up the right framework here would also be useful as a precendent for e.g. Mars; as-and-when we finally stop blowing each other up over lines on a map and set out to explore.