I'd imagine days would just start at different hours, it's pretty ambiguous right now anyway for many applications such as for example the way shifts are calculated.
Before abolishing unified time
I want to call my Uncle Steve in Melbourne. What are the working hours there? Google tells me it is currently 7:00 to 15:00 there. It's probably best not to call right now.
After abolishing unified time
I want to call my Uncle Steve in Melbourne. When are the working hours there? it's 8:00 to 16:00, same as it is here, of course! Same as it is in New York, Bangalore and Hawaii, at the South Pole and on the Moon.
You get the point...
I think you have it backwards: this is an argument for keeping timezones.
If you we get rid of timezones, and everyone lives in UTC, then you already know the time in Melbourne: it's the same as where you live. But that does not tell you the human schedule in Melbourne.
Right now, you can probably guess that business hours are "9-5" (09-17h), and so you can do a TZ lookup to see if people are in the office.
If you get rid of TZs, and the time is 22:15Z, what does that mean elsewhere in the planet? Is that business hours for LA, NY, Paris, Mumbai, Tokyo, or Auckland? Where is that when people people are sleeping?
We know what daylight hours in the local timezone are 8-8 (08-20h), so after doing a Google TZ search we know if someone is likely to be asleep or not.
It's more practical because you can almost instantly learn what hours the sun is up anywhere in the world, which lets you make an educated guess about whether people are awake or in the office (and this would work without timezones). It's less beneficial because you can almost instantly learn what time it is anywhere in the world (with timezones).
Sure, it would be 9am GMT everywhere in the world, but it would still be "middle of the night" or "early morning" or "late evening" just the same. Everyone would just call that 9am GMT.
If you lived in London and wanted to talk to someone in Hong Kong, you'd still need to figure out what time makes sense for both of you. It might be 12pm GMT in London and Hong Kong, but your Hong Kong colleague is fast asleep at 12pm.
People that do care about a clock second being a few microseconds +/- can still do it themselves.
I think the way to transition is to use dual time (local & UTC) in software.
But is is not very practical.
The time zone communicates the part of the day in any place around the world.
UTC has leap seconds. You are thinking of TAI:
Similarly, every time I travel to a city in a different timezone, I'm going to have to find out what usual business hours are, when dinner typically is (which admittedly varies a bit from country to country anyway), and so forth.