I feel like the author completely ignores the simplest answer to wether or not it's ok to call their uncle. Just use the exact same process they would use now. Instead of "It's that time for him" he would think "It's the time of day for him that corresponds to that time for me" but that's all that would change.
Exactly. His point is that people who argue for abolishing time zones are playing a classic shell game. They're talking up the benefits of the new system while ignoring its problems (and the benefits of the old system).
This kind of shell game is pretty popular in software development, where new solutions are hyped because they lack some of the problems of the old solution, but also lack many of the benefits of the old solution.
If the only problem with abolishing time zones is having to look up a table to know when to call someone, then perhaps it's better to abolish them. Most of the uses we make of international times don't have anything to do with timing phone calls.
To make a decision on what's best obviously needs a comprehensive look at all the pros and cons, not just one word-gamey one like this.
Perhaps the current set of time zones are not optimal. Maybe the best solution turns out to be just rearranging them. I would hope making them wider but perhaps there are biological reasons to refine them even further and make daylight savings even more complicated. If that somehow extends out lives, makes us more productive, or saves enough power (money), it might be worth the added cost to programmers (money).
Obviously, that lookup table is not the only problem with abolishing time zones, the giant problem is forcing billions of people to change their notion of when "9 to 5" happens, as well as every other activity that is pinned to a specific time through custom, culture, and language.