The primary issue is when the day changes in the middle of the local day. Assume everyone adopts UTC. Then in California, midnight will happen at 5pm local time.
* Friday UTC 2300 = local time 4pm PST - This is Friday
* Saturday UTC 0100 = local time 6pm PST - Is this Saturday?
* Saturday UTC 2300 = local time 4pm PST next day - This is still Saturday?
When would Saturday afternoon be? We'd have to change our vocabulary, and I don't see that worth it.