Good god, imagine 360 timezones to describe each longitude.
Each timezone would be 4 minutes, and span roughly 56 miles (tho, that's different as you get nearer the poles).
For the majority of things, it would be fine. Most appointments etc that are "booked" verbally would likely be within 56 miles, where "casual" time would work. Anything beyond that feels like a "significant" thing, which would probably involve written/digital communication - where computers could pick up the slack for translation.
And EVERYONE would be aware of timezones. So, even Microsoft/Excel would have to recognise that timezones are a real thing.

So, probably not that bad

close, there's a regional local solar time (you could just steal timezones for this one) and then there is global time, which is what we go by for everything. Local solar time is essentially just an offset to the global time for the relative nature of local time to global time.

i like how whenever i mention this, people seem to think i want to get rid of time instead of timezones