I mean what's more important to you, being able drop the info about time zones when scheduling international meetings, or preserving humanities ability to communicate time respectively to the actual time of day?

I've lived in three countries so far and never actually had trouble scheduling anything. The concept of time of day on the other hand is pretty prevalent in my daily life.

I suspect the language would change for those scenarios, so maybe you would say "I slept until midday" rather than "I slept until 12pm".