Because the Nazi skinheads did a great job of co-opting the punk title

Nah. They tried REAL hard to infiltrate and take over, but real punks always managed to repel them.

for most people who weren't actually interested in punk

And/or were just old fogies.

The news media were very complicit in pushing that specific image to undermine the punk movement to the general public.

Corporate news and undermining left wing people by conflating them with far right hate clubs. Name a more iconic duo.