I've always considered C the speed of light and gravity goes at the speed of light, not that light and gravity both go C, which is a property of spacetime. This is a much simpler mental model, thanks for the simple explanation!
You can think of c as the conversion rate between space and time; then, light (and anything else without mass, such as gravity or gluons) travels at a speed of 1. Everything else travels at a speed of less than 1.
(Physicists will in fact use the c=1 convention when keeping track of the distinction between distance units and time units is not important. A related convention is hbar=1.)
You can tell that c is fundamental, rather than just a property of light, from how it appears in the equations for Lorentz boosts (length contraction and time dilation).
