I do not find that article very persuasive. It compares the ease in the current system of knowing if it is a good time to call your Uncle in Melbourne to the purported difficulty of doing so if we have a single worldwide timezone, but I think it greatly overstates the difficulty.
In fact, you use exactly the same procedure in both systems. It's just that a couple of the values are interpreted differently.
0. In both systems you have to know that Melbourne is about N hours ahead of you due to the longitude difference.
1. Let C1 = your clock time.
2. Let C2 = C1 + N
3. Is it a good time to receive calls AT YOUR LOCATION when your clock time is C2?
4. If yes, call your Uncle.
In the multi-zone system, C2 is the current clock time at your Uncle's location. Clock time approximates local time in this system, so you need to figure out if that local time is a good time to call. In step #3 you take advantage of your knowledge that schedules in local time are approximately the same in Melbourne as they are in most other cities, including yours, so you can use your experience with your local time.
In the single worldwide zone system, it still works. C1 is now the clock time at both your place and your Uncle's place (and everywhere else). C2 is now approximately the clock time when your local time will be what your Uncle's local time is right now. In other words C1 at your Uncle's place is like C2 at your place.
There are good arguments for keeping time zones, but ease of figuring out if calling someone far east or west of you is going to wake them up is not one of them.