Punk rock started in America, with people who were sick of the overproduced music they heard.

Kind of... it was originally used by music critics as a long after-the-fact term for 60s garage-band and surf-rock music, which had a low barrier to entry and a significant DIY element. The 70s musicians who built on those styles were generally much more knowing about their esthetic, and the diversity of styles within the mid-to-late-70s punk-rock genre was wide. Just within the NY scene, you had art-rockers like Television and Patti Smith, retro-garage acts like the Ramones, bands like Suicide incorporating electronic elements, and (soon after) power-pop like Blondie, and the girl-group, dance music and novelty-influenced B52s. And some of those had quite sophisticated production and good musicianship, though sometimes with arftul touches to make it seem raw.

It's also worth noting that the critics were sometimes wrongly ascribing noble-savage qualities to bands that actually had some serious complexity to their music such as the MC5.

A Brit named Malcolm McLaren heard the phrase and ‘invented’ the Sex Pistols band out of his own imagination

He'd also had some prior involvement with the New York Dolls. It's easy to overestimate McLaren's significance in the Sex Pistols, apart from their look and McLaren's undeniable ability to manipulate tabloid outrage. There was already a fashion scene in London that McLaren was able to tap into, and the musicians who could actually play had mainly come up through pub-rock. McLaren's primary strengths were as a trend-chaser and hype merchant. But as his subsequent career showed, he was never a visionary. John Lydon was better at coming up with new ideas than McLaren was, and Lydon was by no means a genius.