Time seems to simply be a way to keep track of matter in space, that is it seems to be a construct to explain the state of matter. Unless matter's state is being undone in space, I can't imagine how time would move backwards.
On the other hand, we already have leap seconds. They're not much of a problem.
Well, leap seconds are going away, so we don't.
They were never adequate for the long-term anyway, as the "two leap seconds a year" scheme will break down as the earth's rotation slows over the centuries and more and more leap seconds will be needed.
We're talking about an hour time drift over a few thousand years. Does that really matter? It's like caring what Julius Caesar's sense of "morning" or "evening" was. They never really solved any practical problem.
Approximations in calendar time can be used for the future where precision timekeeping is not needed (the holiday party next year at 8:00PM), and exact monotonic timestamps can be used where precision is needed (e.g. the timestamp to start a re-entry or orbital insertion burn around Mars).
Our calendars are already merely tags/counters for physical events (solstices, new moons, high noon) we observe, so making that distinction official in timekeeping standards makes sense to me.
Ideally, one would have a monotonically increasing time from some event, but then transforming that into some human readable time is complicated. And of course, what actually is the human readable time? When should 'noon' actually be and should it be adjusted continuously with leap seconds, minutes, etc.?