One can, in fact, regard everything as traveling at c through spacetime. Right now, sitting at my desk, I’m moving at 1 second per second in the positive t direction. If I had some spatial motion, then the component of my speed in the t direction would have to be different, to keep my total speed at c. Ultimately, this is the cause of time dilation in special relativity.
Ring, I seem to recall a quote (probably apocryphal) by one of the great physicists at the turn of the century that the only challenges which remained for physics were explaining the spectrum of the hydrogen atom and the anomalous perihelion procession of Mercury, and that once those two minor details were done, it was all over.