It seems that no amount of mathematics can help some to see what Einstein was saying.
>>>" 'clocks do not measure time...' "
Of course they do. But there is no priveleged frame of reference -- nothing can be separated from time, just as nothing can be separated from space. It's called spacetime for a reason.
>>>" 'I say clocks measure what Einstein calls "proper time".' "
See? Stafford is looking for the luminiferous ether.
>>>" 'The issue is that clocks absolutely always measure proper time along their space time path!' "
This is not a problem! Clocks cannot measure temporal distance until compared with other clocks (remember, it's called relativity). How is it, you may ask, that time -- just like distance -- is a physical function? Because, as Einstein told us, time is a dimension. The metaphysical "time" Stafford seeks was thrown out with the ether a century ago.
Put it this way -- my yardstick always tells me the space between its own "31" and "32" is exactly equal to one inch. Wow! Does that mean my yardstick cannot properly measure spatial distances? According to Stafford's logic, the answer is no.
No meter can measure itself. Everything is relative.
>>>" 'But, is the path length of their trajectory the same as time? Well of course not...' "
Actually, this "path length," in temporal terms, is the same as elapsed time. The space (i.e., distance in spatial terms) covered during this proposed trajectory is usually called distance. And, since everything is relative, by comparing these phenomena with other phenomena, we also arrive at calculations of velocity and time dilation.
It's really very simple.