I’m quite confused as to how you’re actually proposing the time should work. I assume that when we talk about abolishing timezones, we mean that everyone switches to a single standard timezone (and that it still goes from 00:00 - 23:59). Are you saying that you would like:

simplest possible solution.

to give an example, let's say we keep midnight as the date rollover. 00:00 of every day would be the rollover point. The date would change at that point, globally. No matter where you are on the world, the time, and date, is exactly the same. That never changes.

Locally, you would account for this by using offsets, i refer to them as timeoffsets, rather than timezones, or time offset mapping, for completion, which gives you a map to your "local solar time equivalent" Most anything you would need to do would be governed by local solar time, or it's related offset. Work for instance, that's how it already works, nothing would be different there, just the funny number that the clock shows you would change. This is literally just our current timezone system, but inverted.

As for the example i used, probably not a good one. That was 24 hour time with noon as the roll over point, just for demonstration. So the first twelve hours are one day, the latter half are the second. Given how twenty four hour time works. The first 12 hours of the day wouldn't be possible on the day after. Essentially, a good way to think about it, would be that it's like even and odd numbers. You can tell them apart, just by the very existence of them. 15:00 would not be a possible time for the 7th in that example, unless you went back in time. That was just an example of a slip of the tongue type thing. If you were doing anything more serious, you would be planning it better anyway. Noon in that example, local solar noon or not, doesn't matter, as that's arbitrary. The point was just a hypothetical.

Though frankly, i think keeping 00:00 as the roll over makes sense, it's very explicit. Even if it's midday. That's a very explicit time change. DST makes the solar cycle aggressively independent of time throughout the year, in each half of the year, so to some extent, it does with date. Like here in the midwest for example, the summer sunset and winter sunset vary by about 4 hours. Which is a thing that changes twice a year, once a year in the one direction. But twice a year for all intents and purposes. Everybody lives with it perfectly fine. I see no reason that 19:00 being the local solar noon would change anything drastically.

My main point there is that we already wake/sleep at different cycles of the day. On the regular, depending on DST, and season. That doesn't make a huge difference to day to day life. Local solar noon as a concept being noon (more explicitly, 12:00) every day is an entirely arbitrary concept. It's kind of cute and all, but like i said, if you really care about representation for it, you can just rotate your clock. Noon to me just marks the midday point, and the point where the sun is highest in the sky. I don't care about the actual time itself. That means nothing to me.

Oh and while we're at it, standardizing 24hr time would be a good move. 12 hr time is dumb.