It still works. You need to know your correction, but it's just like a computer with a slow clock. UTC doesn't explicitly state it, but it's reference is to a non inertial frame, on the earth's surface. (Rotational effects are far smaller than current clock errors, and transmission time errors)

Time zones are what let you know whether it's the middle of the night when you want to call gram gram in Ireland. Or are you proposing we all abide the same wake/sleep times and ignore the sun entirely? Who gets the good shift on that clock?

Well time were first laid down because of trains. Every would have there clocks set to 12:00 the sun at or close to its highest point. And that was when a horse was the fastest you could ever travel. Then trains move fast and can just go around each other if they are on the same track. So we had to make times more offical so we wouldn't dispatch trains on collision courses.

such as tv programs

People do view TV Channels across timezone boundaries right now.

A lot of folk in the Netherlands and Belgium watch British TV and North America, Australia, Russia etc must have lots of TV channels with audiences spread across more than one timezone.

To say noting of migrant workers/families in various countries who largely shun the local TV channels in favour of satellite/internet TV from their home country. (Indeed the advent of internet TV is making the concept of TV schedules pretty obsolete anyhows)

How do they cope ?