For clarity, both people would need clocks though, correct? The top comment sort of implies you could measure this effect simply with one clock in the room, somehow, but you'd need two.
It works because it assumes swapping the cable has no effect on the delay. But the same cannot be said for your hypothetical atomic clock - creating an electronic switch with matched delay is an obvious challenge, but then, even minor physical movement of the cable will change its delay.
If you think about it, this is true for any time measurement. If you only had one clock, how would you know if it was properly calibrated?