How exactly are we supposed to run a modern society without the concept of time?

You can have a time system based around sunrise.

I mean with most clocks also being internet connected computers it wouldnt be too hard to make a system where you measure time in hours after average sunrise in a tike zone.

That is where I live right now it's sunrise +3.

I'll reschedule the daily 'standup' to SR+3.5, the banks open at SR+3 and close at SR+11.

One major consequence I can see is that now we have north south timezones on top of east west.

This would break in the extreme north or south where there are days without a sunrise.

That's what time zones are. You're just proposing more granular time zones. And none of them can fix the problem that in a lot of places the days are just short in the winter.

Timezones don't account for seasonal changes in sunrise times, and DST is a poor hack that tries and fails. Using local solar time would be an improvement in that regard. On short days you just do less.

Timezones are a reasonable compromise between the desire for people to be up during the day, on one hand, and the need for a modern economy to be able to coordinate people over long distances, on the other. Turning up the dial all the way to the left here is going to cause a lot of problems.