This idea would solve one set of problems (synchronizing events across time zones or figuring out the local time while traveling), but would create a whole new set of problems for long-distance travelers.
Currently people everywhere have a common set of expectations about what time the sun rises, when to eat meals, when to sleep, etc. If you travel to a new region, you change your clock once, and you're instantly slotted in to the local expectations. On the other hand, if we followed the proposal above, travelers would have to do timezone-type math for all these events every day.
Say you travel from California to Japan. What time will the sun come up? Well, at home in California it comes up at 14:00 UTC. California is around 120 degrees W and Japan is around 138 degrees E, so Japan is about 360-258=102 degrees east of California. The sun travels 15 degrees per hour, so events will happen about 7 hours "later" in Japan than in California. Add 7 hours to 14:00 and you can expect the sun to come up around 21:00 UTC. Great. Now what time should you make a lunch appointment with a colleague? Usually in California you have lunch at 20:00 UTC, so add 7 hours, modulus 24 to get 03:00 UTC.
It's a lot of work, but at least you'll know when to catch your flight home without adjusting your clock. You just won't know (without some math) whether that will be the middle of the night, first thing in the morning, etc.
This does not strike me as easier than the current system.