Yeah. I don’t think it would need to be powered on continuously either. If you were motivated enough, you could extract the firmware and check the code to see what leap seconds it was programmed with. But yeah, you could tell “it is at least as old as the last leap second it knows about, and is probably younger than the first one it doesn’t know about”. Can’t say for sure that it wasn’t manufactured after the first leap second it doesn’t know about though by a manufacturer that just didn’t care enough to include the new one.

This probably wouldn’t be the easiest way to tell though, there’s a lot of physical clues in manufacturing methods and whatnot that would likely be easier to date it by.

If the device was a nuclear clock in the same reference frame, maybe.

Maybe, if:

• leap seconds only ever increment (or decrement)
• The accuracy of the device is substantially more than 1s per year, over a long timespan.