Stealing from another commenter: Are you OK with referring to days of the week as Tuesday/Wednesday, or do you propose abandoning day names altogether? If you say your local day is Tuesday which doesn't align with someone else's Tuesday, you've still got the old time-zone problem just at a coarser grain.
As for "secondary time" yes. That's called local time. Which is what the initial proposal was trying to be rid of.
Now riddle me this: What time do you have your computer's motherboard set to?