The biggest problem with that idea is that time would lose much of its meaning.
To give an example, let's say you're taking an international flight that get in at 4:30am. So, will public transportation be running or will you have to get a taxi? Can you call the friends you're visiting when you land or will the be asleep? The problem is that 4:30am has lost its meaning until you contextualize to the location where you'll be. In the current system, 4:30am is almost always pre-dawn and most everyone will be asleep. That's true no matter where in the world you are because every place has its own 4:30am.
There's value in a universal understanding of what a time means. If you removed time zones, you'd quickly see that you'd need to add place codes to every time. This would be because despite the fact that you know the precise point in time represented by an hour:minute value, you'd know almost nothing else.
Exactly. The good thing about the whole time system we have right now is that you can tell exactly what is going on at someone's place when he/she mentions a local time. If it's 3am local time at someone's location, you know he probably won't be at work and that it would be pointless to call them. Yes, you would have to look up the time zone difference to actually know that it is 3am there right now. But if we had the same time everywhere and it's 10am right now, you would STILL have to look up some information to know what the effective difference to that location is, because even though it might be 10am everywhere right now, that person won't be at work if it is still in the middle of the night. So there would STILL be "time zones", they just would not be called that anymore. So you'd have 10am universal time, but they might be at -7 "working time" difference. What would be the whole point of the universal time then?
You just cannot have an effective universal time as long as it is not night/day at the same time everywhere, and I do not think we are close to fixing that problem anytime soon, because it might have to involve transforming our planet into a flat earth.