> my key point there was simply to show that the magnetic field isn't really necessary - I wanted to show that it's all part of relativistic contractions made by the electric field.

This isn't quite right, there are field configurations where the magnetic field doesn't vanish in any reference frame. This is actually the typical case: consider, for instance, two point charges moving relative to one another.

The right takeaway from SR isn't that the magnetic field is fake and the electric field is real, it's that both magnetic and electric fields are frame-dependent and it's the electromagnetic field tensor that's the real physical object.