Wouldn't this net positive charge attract the electrons moving through the wire?
According to the electrons, yes.
According to the ions, no.
Also, isn't this what a magnetic field due to moving charges is? (electric field from a different frame of reference). Or is that different.
Those moving electrons are currents. Those currents attract each other, as we know from old textbooks about magnetism. This attraction is equivalent to the repulsive Coulomb forces between the electrons being reduced. Those repulsive Coulomb forces are the forces that are preventing the electrons from being pulled to the center of the wire by the positive charges. I encourage the reader to think what happens when the forces that are preventing the electrons from being pulled to the center of the wire decrease.
Note carefully that the pull of positive charges has not changed. (Except if we ask electrons, they would say that the pull has increased. We can explain why electrons feel that the pull has increased by the effects that the motion of the electron has on the electron's views or opinions or measurements, relativity considers just that kind off stuff all the time)