electromagnetism - Is magnetism an electrostatic or electrodynamic effect in the rest frame of the affected charge?

In textbooks, such as[1,2], magnetism is taught to be a consequence of relativistic length contraction. The magnetic force is derived in the special case of an infinite straight wire, by finding the total line charge density $\lambda_{tot}=\lambda_++\lambda_-$, and evaluation it as if it was a static charge. The Wikipedia page on the subject[3] states that "The chosen reference frame determines if an electromagnetic phenomenon is viewed as an effect of electrostatics or magnetism". I have found that this is not correct. The textbook analysis will only work in the special case of parallel motion next to an infinite wire.

This can easily be seen by considering perpendicular motion of the test charged with respect to the infinite wire. If we let the test charge move along the x-axis, and the charged wire run along the y-axis, the Lorentz transformations will be independent of y. All charges in the wire will have the same x' and t' coordinate, and there will be no length contraction along the y-axis. The line charge density will not change. $\gamma_-=\gamma'_-$ and $\gamma_+=\gamma'_+$. Hence the wire is electrostatically neutral in the rest frame of the test charge, and no force occurs.

The problem can be resolved by using the dynamic (retarded) electric field in both frames. (eq. 1) See my in-depth analysis in this link: https://drive.google.com/file/d/1HITikNdOX-IbxHmQVZVKQLATOrNXheXp/view?usp=sharing

$$E_D=\frac{q(1-v^2/c^2 )}{4 \pi_0r^2 (1-v^2/c^2 sin^2 ( \theta ))^{(3/2)}} \mathbf{\hat{r}} \tag{1}$$

I come to the conclusion that magnetism must be understood as an electrodynamic phenomenon in the rest frame of the test particle, and as a combined effect of field retardation and relativistic length contraction.

Question: From my analysis, it seems obvious that the magnetism is an electrodynamic effect, and yet I have found no mention of it in either textbooks or on the internet. There seems to be a common agreement that it is an electrostatic effect.

Am I missing something here, or is there really a wide spread misconception about this?

References
1 R. Feynman, The Feynman lectures on physics volume II, chapter 13.6

https://www.feynmanlectures.caltech.edu/II_13.html
2 David J. Griffiths, Introduction to electrodynamics, third edition, chap. 12.3.1
[3] Wikipedia: Classical electromagnetism and special relativity
https://en.wikipedia.org/wiki/Classical_electromagnetism_and_special_relativity#Relationship_between_electricity_and_magnetism