The old saw is, "measure with a micrometer, mark with chalk, cut with an axe."

Using GPS to provide a source of time signaling directly through human senses is always going to be swatting a fly with a hydrogen bomb. Humans just aren't very accurate preceptors without lots of technological assistance. The visual GPS clock that presaged this project wound up with an accuracy of ~180 µs, but the granularity is only 100 ms. The speed of light has essentially no bearing on the accuracy.

Unfortunately, I don't have any information about the latency of human perception.

The talking clock has a granularity of only 1 second. But how accurate is it?

Because the tick/beeps of the talking clock are 1 kHz and the oscillator for them is free-running, the accuracy spec starts at ±1 ms, but there's also the PPS ISR's latency to consider, which is probably a few microseconds (that ISR has little to do other than enable the tone output). Depending on where you are in the room relative to the clock will also have an impact because of the speed of sound - probably adding another 2-3 ms of latency if you're ~1 meter away.