There is a gap in reasoning here. Prior to timezones, all localities had their own time, and it was generally based on solar time. Thus, abolishing time zones

There is a gap in reasoning here. Prior to timezones, all localities had their own time, and it was generally based on solar time. Thus, abolishing time zones could mean going back to local solar time, but the article presumes that this means going to UTC. (Whether it is UT1 or UT2, it doesn't say). This isn't all that complicated. Pick some astronomically-based event, say when the sun is at the zenith on equinox, and tie local time to that. Then, given that you have coordinates for every significant place on earth, it is an easy computer calculation to calculate the delta of time to where your uncle was sleeping. Similarly, the offset to UTC is trivially available. This is a calculation that ham radio operators, among others, already use to know what UTC the sunset or sunrise is (these are interesting times due to ionosphere changes around those times) at some random distant place far away, say Vanuatu.