Your sentence would have more merit IMO when rearranged. No, but metal has influenced most all punk made since the late 70s.
My point is, that as a matter of musical theory, there are elements of heavy metal that define it as a genre, that are not present in punk rock.
Regarding the 'punk vs metal' discussion...
I think the question is mostly moot. I believe Punk was a direct, cynical attack on the music industry. Elements of Metal? Most definitely - with some groups later on. But going back to the earliest examples - New York Dolls, Devo, and (later) - Sex Pistols - the sound was the _evolution_ of REJECTED 60s counter-culture music. So was Metal and many other genres.
Pop, Beatles, Disco, all became the beloved music. When 1980 hit, and the US and Overseas public vomited up all the corporate music, New Wave was an attempt to darken that vomit into something new.
I would propose that Metal and Punk were siblings. Both the unloved children of the 60s.
Metal and Punk shared their thoughts and musical themes, with different musical expression- siblings with common parentage.
New Wave created a bright, happy, albeit different, future - one of a equitable, free (in appearance), techno Futurist (e.g. Reddit).
Punk knows the lie. Deconstructs, refuses to worship 'art' or the technology.
Screams and curses at the abyss opening beneath its feet.
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