@neologist,
I am supposing, Neo, that the spirit of your question related to the
concept of time, not with our method of measuring it, per se. Put another way, your question seems to be one about the nature of time, not the mechanics of clocks.
Inferring evaluative conclusions from purely factual premises is a process that has been called "the naturalistic fallacy." It seems to me that this fallacy is committed when someone argues, for example, that because clocks slow down with speed, that means time cannot be "absolute" (in Newton's sense of time which "flows uniformly"). This simply binds the meaning of time to the mechanics of clocks. I don't see how the fact that clocks change rates with speed can, in itself, alter the
concept of time.
The fact that a clock can "slow down" does not mean that "time itself" slows down. The fact that you haven't recently wound your wristwatch (causing it to "run slow') alters nothing about the "real time." At that point, the time is simply NOT what your watch tells you it is. If it's now an hour slow, and reads 3:00, it still 4:00.