If the photons were travelling faster than the local speed of light, why would light be emitted? The Cherenkov radiation takes away the excess energy as the light is immediately slowed as it moves from whatever radioactive metal to water

Gravity waves are indeed different. I don’t know the maths of relativity, but I bet those equations also require specific wave speeds. They don’t care about matter aside from it’s gravity which can bend space-time and change the wave path

I think we’re pushing this beyond both our understanding

Measuring causality/c despite those being given by our maths refines our values for the various constants