Without timezones, everybody who isn't lucky enough to live near the new meridian will have the calendar date change at some point in the middle of their waking hours, which is a great recipe for confusion.
It'd also mean that in those places, anything defined on a day level (like holidays) would either have to start and end right in the middle of the solar day, or you'd have to start specifying starting and ending hours, too.
And if a place then starts to define standard starting/ending hours which approximate the former local midnight, you basically end up reintroducing time zones through the back door.
Let's also do decimal time while we're at it, because nobody cares for base60 anyway.
My argument for NOT changing anything is the amount of assumptions that have been built into all sorts of critical and non-critical software, it'll simply be too expensive at this point, not to say dangerous.
Timezones exist to serve the purpose of interacting with other areas in relative daytime using discrete chunks of geography.