We're inundated with time displays. Right now I have 5 time displays within 20 feet (computer, stove, microwave, phone, watch). I could imagine some people would continue to reference sundown / sunup for relevant situations (kids, get home by dark) the same as they already do, but not for anything time sensitive (like say a meeting). There's just no way I would schedule a call for "3 hours past sunset" even if I was forced to tell time in binary.
Also, think of how deleterious the abolition of local time would have on communication. Right now I can say something like "I received an urgent call at 3am" and you know immediately what that means. But if I said "I received an urgent call at 17:00Z", a lot of the meaning is lost. You'd have to know where I live, i.e. what my local time zone is, and then do some quick mental math to determine what actual time of day 17:00Z means for me. With local time, that calculation is already done for you! Local time is just too damn useful of a concept. It really truly is better than global time for most uses. Global time is really only useful for scheduling global meetings and computer stuff. Even within the US, all of our scheduling is done by US timezones.
Given your example we must be talking about different things. I'm saying ONE time globally.
[Well], think of how deleterious the abolition of [global] time would have on communication. Right now I can say something like "I received an urgent call at [17:00Z]" and you know immediately what that means [in reference to everything else happening in the world]. But if I said "I received an urgent call at [3am]", a lot of the meaning is lost. You'd have to know where I live, i.e. what my local time zone is, [where the caller is] and then do some [potentially complicated] math to determine what actual time 3am means. With [global] time, that calculation is already done for you! [Global] time is just too damn useful of a concept. It really truly is better than [local] time for most uses. [Local] time is really only useful for scheduling [local] meetings and [in-person] stuff.
Also, I don't understand what point you're making. The square bracket stuff you've added doesn't work. You haven't managed to correctly communicate the fact that the person was woken up in the middle of the night. Which is what local time is extremely good at and global time cannot do -- putting a specific time in context with the rhythms of the day. Which, you know, is very important for most normal communication. I can't even schedule a worldwide meeting using global time; I have to use the local time of each participant individually to figure out what the best time is that maximizes the # of people calling in during the workday and minimizes the # of people that need to be up in the middle of their local night.
Is Tokyo in the some timezone as Beijing? Is London in the same timezone as Reykjavik? Did Mexico start daylight savings last week or now? I have to look up stuff to answer these things, as well as the local timezone designation (is it ET or EDT? Is it CT stand for "california time" or CT for "central time"? Is there a CT in another part of the world that could be misunderstood by another participant?), so that I can publish the meeting time correctly without people misunderstanding it. Roughly avoiding unreasonable times is much easier to do than this. The sleep times of Reykjavik and the sleep times of London don't really differ by much, so as long as the proposed time steers clear of that, it will be fine.
In fact all I need is a world map that shows me the day/night part of the Earth as I slide the (UTC) time -- there are many apps that do this already. Then schedule the meeting such that the greatest number of participants fall under the daylight. Then publish the meeting as a single UTC time. That's it.
If somebody wanted to communicate that they were woken up in the middle of the night, they could use this perfectly fine sentence:
"I was woken up in the middle of the night."
Communication wouldn't break down just because everybody didn't have an identical reference point w/r/t timestamps in relation to daylight cycle. Something we don't have today anyway, by the way. When is dinner, for example? (conservative answer: 16:00 to 23:00).
Local time is incredibly useful. It's never going away. It's utter fantasy to think that everyone is ever going to just give up local time and only speak in vague terms like "an hour after noon".