we have 200 years of photographic, and now digital pictures of the night sky. The solar system moves 828,000 km/h(230 km/s) this is an incredible speed.
So we have photographic images of the night sky, which are millions of miles apart. Which allows us to triangulate stars, and work out their speeds. See https://en.wikipedia.org/wiki/Stellar_triangulation
Red shift is caused by interstellar antimatter, giving light a slight red shift. The further we look the more red shift there is.
It so obviously has nothing to do with relative speed. Triangulate the stars, and get their real relative velocities.