Left/right, port/starboard, north/south are all in common use because we need to be able to talk about directions relative to different frames of reference. It’s just as true that we need to talk about time in multiple different frames of reference.
It still seems less complex than tracking dates across time zones.
Having it be the next day when you wake up is really intuitive and kinda practical. Having 9.00 mean "some time during the morning" all around the world is also quite comfortable.
I think from the perspective of a travelling human time zones are more practical than not having time zones, especially since your modern devices automagically change them anyways. Such an "automagic" change could not happen when 9.00 suddenly means "middle of the night".
For non-traveling folks that coordinate via the net it could surely have it's merrits (which is why we have UTC on servers?), but even there I could imagine software could help.
E.g. calender tools could just have a tool that allows scheduling for international participants. You enter the locations and it shows you the different day cycles next to each other — because after all isn't this the issue? Finding a time where it is roughly day-like for all participants of the event?
Using a universal time zone wouldn't really help you there, because you would still need to know what 9.00 means in some other part of the world.
Isn't that in some ways just reintroducing time zones through the back doors?