This is going to be a long post since there are multiple issues being revealed by this question.
1) alternativephysics.org is a bad site
I've responded to another part of that site in another question. The writer misunderstands relativity and comes to bad conclusions from that misunderstanding to "prove" that relativity contradicts itself. Then, the writer proceeds insult anyone who actually understands relativity instead of finding out what they think. I'm not going to write anymore about them.
2) The length-contraction argument is a pedagogical tool, not a derivation
Sometimes, instead of a rigorous, math-heavy derivation, an intuitive and simplified argument is given to students to convince them of the real relationship between multiple concepts. For example, the following shows the equivalence of Kepler's third law of planetary motion and Newton's inverse square law of gravity:
r^3 = kT^2 &\iff r^3 = k(2\pi r/v)^2 \\
&\iff r = 4\pi^2k/v^2 \\
&\iff v^2/r = 4\pi^2k/r^2 \\
&\iff mv^2/r = 4\pi^2 km/r^2 \\
&\iff F = 4\pi^2 km/r^2
where $r$ is the radius of the orbit, $T$ is the period of the orbit, $k$ is some constant, $v$ is the constant speed of the orbit, $m$ is the mass of the orbitting body, and $F$ is the gravitational force. However, this only works for circular orbits. The full derivation of this equivalence for elliptical orbits takes an entire chapter of a graduate-level text book.
Likewise, the length-contraction argument only works when the test particle (the one that feels the magnetic force) moves at the same velocity as the flowing electrons in the wire. That restriction makes the argument much simpler, which is good for convincing students that electric and magnetic fields are the same thing from different points of view (a.k.a., reference frames). But, that same restriction also restricts its generality. In fact, using the same argument, you could conclude that a positive test charge at rest with respect to the nuclei in the wire should be attracted to the wire with a current since the electron spacing should be length-contracted, making the electron density appear greater than the nuclei density. This is obviously not the case.
For a better derivation, I've found this (non-peer-reviewed) paper that uses the relativity of simultaneity to derive the magnetic force and the lack of force on a non-moving test charge. In this paper, the test charge is not restricted to moving at the same velocity as the current in the wire. The math is more difficult and even makes reference to a textbook simply referred to as Jackson--a tome much feared amongst physics grad students (me included). However, the first two sections should be manageable for most people familiar with special relativity.
In conclusion, the length-contraction argument serves a purpose: to relate electric and magnetic fields in different reference frames to students who are first encountering these ideas. It is not rigorous or general, but it is useful.