We used to have that, and it was dreadful. Every municipality had its own local time. Railroads made that untenable, and led to the creation of timezones.

Is it possible that such a system would be more tenable now with cheaper and more ubiquitous devices that can measure or calculate local solar time?

Well, railroads still exist.

Two people viewing the sun at its zenith at the same time can disagree as to what time that occurred by hours using standard time. You’re advocating the abolition of time zones...

They didn't say not to round it!

For non-distorted time zones, when the clock strikes 12 everyone can look up and say that the sun is roughly closest to 12.

Yes, that's the point. Of course it should be sensible and rounded.

If we were too literal with this definition, 12 would even change everyday because of the analemma [1].

[1] https://en.m.wikipedia.org/wiki/Analemma