This is pretty much false. For one thing, they can’t all have the same velocity relative to each other because each of them would be moving in a different direction, due to being at different points in the Earth’s orbit. For another, they’re unlikely to have the same speed.
Also I’m not exactly sure what you meant, but even if they had the same velocity relative to each other (which they don’t), it wouldn’t follow from there that you could ignore the effects of relativity. This is a triangulation problem from 3 or 4 satellites to a receiver on Earth. Even if all the satellite clocks were off be the same amount from Relativity, the position indicated by them would still be off, though the true position could be systematically worked out. That said, they are not off by the same amount, as they move at different speeds. You could draw a diagram and work this out straightforwardly.
Not to mention that relativity clearly affects at least the implementation of GPS, since it is known that the GPS satellites are, in fact, corrected for relativistic effects.