> If this were applied to the galaxies spiral arms, then as they continue to orbit, the inside ones would orbit faster than the outside ones, and the galaxy would become increasingly spirally. By this point, the galaxy would be so tightly wound that we would not even be able to see the spiral any more.

I thought that problem was the same as the galaxy rotation curve problem? If we solve (erm) that issue by adding in some dark matter - can't we then also accept a spiral structure to rotate with the galaxy without "winding up" the spiral?