Yes, this is how I learned it. Suppose you have a line of charges with density p moving up at velocity v, and want to find the force on a stationary particle with charge q and distance r to the right. The classical Gauss' law gives

F = (p / 2πrε) * q

If we switch to a moving reference frame (up at velocity v), the charge density decreases to

p' = p*sqrt(1-v^2/c^2) = p - pv^2/2c^2 + O(v^4/c^4) (from Taylor series)

The force should stay the same, so

(p / 2πrε) * q = (p' / 2πrε) * q + F'

where F' is some other magnetic force created by the current. Solving, to second-order we have

F' = pqv^2/4πrεc^2

If we introduce a new constant

µ = 1 / εc^2,

we get

F' = (µ / 4π) * pqv^2 / r

Now, we know

F' = B * qv

where B is the magnetic field, so

B = dF'/d(qv) = µpv / 2πr = µI / 2πr (where I is the current in the wire).

This agrees with the Biot-Savart Law for an infinitely long wire.

It's correct, but I think it's misleading. If you're now thinking that magnetic fields are just electric fields viewed from different reference frame and that the latter are the more fundamental, that's not right.

There are situations in which you have both fields and you can't attribute the magnetic field to a purely electrostatic field in another frame. It turns out that all observers agree on the value of k=E²/c² - B², so if you have k<0, you can't possibly find a frame with no magnetic fields, because that would imply k≥0.

So, you can use relativity to motivate the need for introducing a B field, instead of "pulling it out of the hat", but in general you need both E and B. In the modern formalist these are the components of a larger object called F, the field strength tensor.

The article makes it sound as if magnetism arises because of a delay in the propagation of the electric field caused by a limit on the speed of light. It's kinda the other way around. Light is a alternating electric field and the speed of it's propagation is limited by the ε0 and μ0 (electric permittivity and magnetic permeability) in Maxwell's equations. From Maxwell you can derive c^2=1/(ε0*μ0). Lorentz and Einstein came along and realized that this changes all of mechanics, and that is special relativity.