That's greatly exaggerating the difficulty of deciding if it is OK to call your Uncle who is about 165° east of you in a single time zone system compared to the current multiple time zone system.
In fact, the procedure is the same in both systems:
1. Look at your clock.
2. Add 11 hours to clock time.
3. Ask yourself if it would be daytime at your location when your clock says that time.
4. If yes, call your Uncle.
• Under the current multiple time zone system that works because clock time is close to local time. Step #2 computes the clock time at your Uncle's location, which gets you close to the local time at his location.
Daylight hours are approximately the same in his local time as in your local time, aside from the seasonal variation which you can ignore if you allow yourself a reasonable margin of error (e.g., aim for calling him a couple hours or so after sunrise rather than shooting for the crack of dawn), so you can use your local time as a proxy for his [1].
• Under a single worldwide time zone system it works because your local time is about 11 hours behind his local time. The time you get in step #3 is what the time will be on your clock (and on his clock, and on everyone else's clock) when your local time is the same as his local time is right now. So if that clock time will be a good time for you to receive a call, now is a good time to try your Uncle.
Of course, you probably wouldn't actually have to do any of that. Just as in his first example with the current time zone system where he Googles "time in melbourne" rather than looking at his clock and adding 11 hours, in the single worldwide time zone you would Google "local time in melbourne". In such a world, where local time can be quite a ways off from clock time, Google would surely recognize such queries and do the right thing.
Heck, it might even assume you mean local time if you just did "time in melbourne" because in the single time zone there would be no need to Google for the clock time for a particular area. I'd expect "time in melbourne" to probably give something like "The local time in Melbourne is X at world time Y". That covers everything.
[1] If you really are aiming for crack of dawn calling, then you will have to take into account the seasonable variation in sunrise relative to local noon, but you have to do that in both the multiple time zone system and the single worldwide time zone system.