Right now, you understand the basic meaning of different times no matter where you are: Noon means lunchtime, 8am is rather early, 10pm rather late etc. You can travel and this doesn't change. You can watch British movies and you'll understand the time references.
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.
It sounds like a decent idea on the surface but then times would be meaningless. Right now a time "means" the same thing everywhere. When we ask "what time is it in Japan?" we understand the answer. We understand when its "noon" in Japan, we know whereabouts in the day it is and what we can expect people to normally be doing, ex, eating lunch. If it's 3am in Japan we know its not a great time to call beyond emergencies.
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.
An alternative is just to count seconds. Be as precise as you need (m? k? M?). A little like unix timestamps, except without the rule that says "there's 86400 of them in a day no matter how many seconds there were in that day".
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"?
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.