I think maybe we're just measuring distance wrong. If we use light-based measures of distance, of course the speed of light is going to be constant. Really distance between A and B is more accurately described by work required to scale the potential difference between A and B. It's like you have two points that are attached by strings and pulling one taut distorts the other. Neither string is innately distorted, but choosing one to pull taut distorts the other. It's our choice of light-based inverse square measure that makes spacetime look warped.
Maybe this is what relativity is saying, I dunno, too much of a brainlet.