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