That is, I like the idea in principle, but I know it'll never happen.
If you're going to argue that Google can solve the problem of "what time is it over there" in our current system, then Google can solve the problem of "what times are office hours over there" too.
I do have to admit, it did not occur to me that the work week would still effectively enforce an "ordering" on longitudes which somehow takes the beauty out of it.
You know what's beautiful? The fact that the vast majority of humanity is awake, engaged in their major activities (school, work or leisure) at 12:00, no matter where they are in the world, and that this is a shared point of reference.
It's not an exaggeration to say that timekeeping is at base, about the daily cycle of day and night. Every development we have created in the past four millenia is an iteration on that same idea. There is day, and night, and our clocks are just more precise ways of marking that cycle. You want to abolish that, for what? A nicer API?