Steve,
If the Earth was alone in space and not rotating at all, GPS would still work just fine.
The leap second is needed because Earth's rotation is slowing making our days slightly longer by the sun.
GPS satellites each have their own atomic clocks and do not care when the sun or a star crosses anyone's meridian.

I am quite sure that Dr L knows all that very well and was just being flippant, which is what panellists do on that programme.