Nonsense. Photons go along trajectories in spacetime just as much as anything else. It's the length of that trajectory that's so special to them.
Learn special relativity on its own terms, then criticize it based on its ability to deal with experiments. You're never going to get anywhere by arguments from incredulity, because STR is just as mathematically consistent as Euclidean geometry.
Velocity is orientation in spacetime; the fact that going near the speed of light mixes space and time measurements is no more mysterious than turning around in space mixes my "front-back" and "left-right" direction. Since velocity is an orientation, it can be described by an angle to some observer's time axis. The relativistic velocity addition formula just says to add the angles. It's conceptually straightforward; if restricted to one spatial dimension and time, it's even no more complicated than coordinate geometry on a plane that grade-schoolers learn. Until you get to dynamics, anyway.