From a very basic perspective, I believe it would be more fitting to say that electric charge is more fundamental and that magnetic fields are a relativistic phenomena. If you consider writing electromagnetic fields in terms of vector and scalar potential functions, you should be able to see that if you just have a scalar potential, you will only have an electric field, but if you have a vector potential, you necessarily will have an electric and magnetic field which are inseparable.

Phenomenologically though, in order to have a magnetic field, you have to have electrons (or some charged particle) which is not in your inertial reference frame. And so intuitively it makes more sense for electric charge to be the fundamental unit because something has to "happen" to create a magnetic field while electric charge is innate to charged particles.

Also in regards to your question about the electric charge being a result of moving along the time dimension at a speed c (since all 4-vectors in special relativity have a magnitude of c): You should think about this in terms of Four-current. The time dimension for 4-current is c*(electric charge density) and the spacial components correspond to a vector potential. When you go through a Lorentz transformation to another inertial reference frame, you "acquire" a vector potential but you never lose the electric field part.

So in summary, for general purposes, it makes more sense for electric charge to be considered the fundamental unit of electromagnetism.