I feel like days are a non-issue; they would just start at different times (UTC) in different territories. This wouldn't make things any more complicated than they already are (currently, if I want to talk to someone in Australia, I have to look up what time it is in Australia and infer the day of the week from that, if necessary. If everything is under UTC, I know what "time" it is, but I still have to look up what day it is).

Most of the issues time zones cause are not "day of the week" related anyways (at least in my experience), so I think having to figure out what day of the week it is somewhere else wouldn't be a common problem anyways.

There's certainly a bit of dramatization/exaggeration here, but the main point is that it doesn't really fix the stated problem while also being a huge change for everyone to adapt to.

I think you missed entire point of operation.

If everywhere runs on UTC, they will still have different times when people are working/not working/sleeping so you still have to look something up and figure it out.

With time zones, you look up "What time is it?", realize it's 4:30AM and since most people around the world follow similar schedule, you quickly realize he's fast asleep.