Bare in mind that just because Uncle Steve is awake that it is not necessarily convenient for him. In the scenario he likes to stay in bed a bit later. This is no problem if you simply send him a message. No need for any lookups or actuarial tables or astronomical calculations whatsoever. If we want to voice chat we can do so after he has responded to the asynchronous chat message.
Bare in mind that the answer to "what time in it" post timezones would be "mid-morning", or "late evening" or "noon". ie. conveying the information you actually need.
However I also think that 'wall clocks' should be something more like the wizard time peaces in the various harry-potter movies.
They'd all have fuzzy times like "dawn, noon, dusk". What is now 6am would be "early morning", 3 hours later 'morning'. Evening would be 1800 and late evening 2100. The times closer to mid-day would be on a denser schedule (finer granularity).
If we all used UTC around the world, I'd have to calculate the difference for all the times that I'm used to. In London the stores open at 9 in the morning, but if I travel East a bit into Europe they open at 12, and dinner is at 11.
It has nothing to do with jet lag. Currently you only have to do any time calculation is you're working across time zones, once you're there and dealing with things in that time zone there's no real difference from what you're used to.
It doesn't matter what number my clock says at noon/sunrise/sunset. If it is about working-hours coinciding with daylight, businesses can adjust their working hours as needed (and post them on their site without you having to convert to local time to know when you can call)
edit: the link below just changed my mind. dammit, I hate being wrong.