Why don't we have one timezone covering the whole earth?
And instead changing the time work and other things happens depending on where you are. Would be easier to arrange meetings across the globe. Same thing applies to summertime. You may start work earlier if you want, but dont change the clocks!
What you do is you have both, kind of like we already do, but with the global time being the default rather than local time. So, if I were to look at my phone right now, it would say something like 1433 9:33AM.
When referencing the time to people I know to be local, I'd use the local time, but any time confusion could occur, I'd use the global time. We have everything in place already, we just need people to get used to knowing what time it is UTC
I doubt most people would use local time in their day-to-day life if global time is the default. You would just get used to the new schedule the same way that you have gotten used to the current one based on local time.
I do think that it might be useful to have something like a "world clock" when traveling. So your clock may say "14:33, like 09:33 at home". But I'm not even convinced how useful this would be. Once you remember one or two timeframe references or if you can see the sun you will have a rough idea of what time-of-day it is anyways. And generally the local schedule will vary a bit from your home schedule anyways so having exact local-equivalent time will probably not be that valuable.