Switching to UTC changes all that. And, almost as important, it doesn't help in the least: you still need to think about your co-worker's geographic location when trying to find a time that's good for your phone call.
If you're in the East Coast US and its 5am and at the same time its also 5am in Japan, but they mean different things, you no longer "understand" what time it is in Japan. You don't really have much of a way of answering "is it a good time to call someone in Japan?" - You have to figure out a way to "translate" from your time to theirs. It would be very, very confusing. So in the end you'd end up re-inventing time zones.
It only works if nobody knows anything about other parts of the world.
And then let "Sunday", "Monday" etc become simply approximate terms. And I never have any idea what day of the month it is so I don't care. (Birthdays and public holidays are set based some recurring feature of the count.)
A refinement of this uses a time period that is approximately 100,000th of a day so then people can easily notice the day. In practice we might use hectokiloperiods instead of periods. Someone will invite a less woeful name.
Nothing about telling time will be particular familiar. Doesn't mean it won't work. But I suspect it'll be awful. I'm fine with daylight savings and all the rest of it. Except days of months. Can't we just say "Monday, third week of June"? or "Wednesday, 39th week"?
We would basically be replacing standard timezone conversion with some kind of non numeric system of identifying the time of day in a place by comparing it to normal activities. We would be getting rid of timezones and create something like "activity" zones. It would be a nightmare.
If sunlight had no bearing on human activity, and people didn't tend to follow similar patterns, then utc would be fine.