A practical solution. As long as we can keep 12:00 near high noon and 00:00 near "high midnight(?)", that should be good enough.

We don't even keep very close to that as it is. Between daylight summer time changes and some time zones already being very far off from geography, some places like western Spain or China are already as much as 3 hours off from actual solar time.

Solar noon already varies by up to 30 minutes throughout the year, due to axis tilt and elliptical orbit.