Skinhead rock was a working class English reaction to punk becoming "too commercial" that started when punk was just barely becoming a mainstream success, and white nationalist and eventually neo-Nazi skinheads arose pretty organically from that scene. Skrewdriver is a good example of this paradigm. They didn't worm their way into the scene one day and try to turn it Nazi. They started as a street punk band that wasn't particularly racist and becoming more and more white nationalist and eventually national socialist as time went on.
This parallels skinhead's evolution as a subculture as a whole. It started as a black and white working class English thing, that ironically drew roots from Caribbean as well as British culture and had reggae as its musical genre of choice. Even in the early days there were racist undercurrents to some portions of skinhead culture... but not against black people. Blacks and whites would both engage in "Paki bashing" because they disliked Pakistani immigrants for taking their jobs.
Then right around the time punk was becoming a thing, "black liberation" and pan-African nationalism also started becoming a thing, and this drove a wedge between a lot of black and white skinheads. A lot of black skinheads abandoned the scene to become kangz (to use a modern term that roughly captures what was going on and why it was so divisive), and a lot of white skinheads started replacing the more overtly "black" aspects of the subculture with alternatives, including replacing reggae with punk. This was by no means universal, and there were still quite a few black skinheads (and still are today) and white skinheads who listened to reggae, but it did produce a noticeable cultural shift and punk completely supplanted reggae as "skinhead music."
Skinhead subculture becoming more "white" led to more white racists becoming a part of it, but the media also contributed heavily to this by biased negative reporting and moral panic shitstirring. As "skinhead" became synonymous with "Neo-Nazi" in the eyes of much of the public, it unsurprisingly made white racists want to be a part of the skinhead subculture, and repelled most other people who weren't already acquainted with it. A negative feedback loop. Then you had William Luthor Pierce and his record label, which is probably about as close as you got to Nazis trying to deliberately subvert punk for their purposes, but Pierce was only exploiting something that had already sprang up without any outside influence.
Skinhead subculture and skinhead rock still aren't, and never have been an exclusively "Nazi" thing. There's the white nationalist and neo-Nazi wing, the "anti-racist" wing which has been pretty much overrun with far left Antifa types, and the "neutral" wing of people who mostly just want to listen to their music, wear their cloths, and bond over working class travails. There are also a lot of people in the "neutral" wing who are more traditionally conservative or libertarian, and a lot of those people get smeared as neo-Nazis by Antifa types, which is very easy to do because most people outside the scene only know skinheads from movies like American History X.
Early punk's affiliation with the left also tends to be exaggerated nowadays. The Ramones had right wing Johnny, and while the Sex Pistols could be considered "anti-conservative" they were general provocateurs, not any kind of left wing ideologues. Band members wore swastika shirts to offend people. And now Johnny Rotten is a Trump supporter who's calling for an end to lockdowns. You also had bands like Agnostic Front that were fairly apolitical but occasionally dipped into messages that were a bit more right leaning. If anything, punk as a genre has been subverted more into an exclusively left wing thing as it became trendy, commercialized, and "radical chic" whereas in the earlier days it was all over the map.
I look forward to Michael "Lords of Chaos" Moynihan himself reviewing this collection at Reason.com, LMAO. 
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