If you’re asking about engine performance then no, the material doesn’t really play into it.
Overall exhaust system design (4-into-1, tri-Y, etc) Tube diameter, runner length, bend radius, collector geometry, port/flange matching, baffle design (if any), thermal coatings and internal surface finish are significant factors.
Full race headers are usually made of mild steel, mostly because it’s easy to form into the desired shape, easy to weld and reasonably priced. Race cars don’t see that much actual run time so corrosion resistance and environmental exposure aren’t issues. (Turbo manifolds have extra considerations due to extreme temperatures.)
Inox exhausts should be great on a street car and very cool, although overkill, on a race car.
PC.