The probable reason people resist this is that local time has information built into it that is valuable. You can tell instantly whether an office is open if you know that local time is, say, 10 am. If you say that local time is 21:00 GMT you lose that information.
Is that more valuable or more convenient than being able to say “the conference call is 21:00” and have everybody in offices around the world know instantly when that is? Hard to say. Maybe in some future time, day and night won’t be an issue. Today they still are.
Presumably after a while of using one particular set of time labels, people would get used to them just as they’ve gotten used to the current ones. “10am” tells you nothing about whether something’s going to be open or not; you need to know what its operating hours are to be absolutely sure. Just because something is “an office” doesn’t mean they follow banker’s hours.
The main advantage the current system has is that the numbers on the clock vaguely correspond to the position of the sun given the date and latitude, regardless of longitude. That is, if you call someone and they say “it’s 6am here!” you know that’s kinda early, whereas if you were all using GMT they’d have to explain they were just getting up/still in bed/whatever.