I'm not sure how this is being construed as a zero sum game where bands can either be influenced by 'punk' or they can be influenced by classical/jazz/folk/whatever, but they can't possibly be influenced by both. It should be obvious that jazz, classical and folk were all significant influences on metal from the get go, given the frequent allusions to all three in the Ur-metal band, Black Sabbath. It should be equally obvious that later metal owes a good deal of its riff lexicon and most of its rhythmic punch to punk, crust and hardcore.
This too, was true almost from punk's inception: can anyone honestly say that "Exciter" would have sounded anything like the song we know and love if punk hadn't burst into the popular consciousness just a few months before? And we're talking JUDAS PRIEST here, not some crusty underdog from the lower tier of speed metal.
On the flipside, the very same album includes "Beyond the Realms of Death," a song that not only includes severally overtly neoclassical gestures, but is very much an electrified version of a sort of British folk music that ultimately derives from the tradition of formal court ballads (usually tragic) of Tudor and Stuart England.
And really, what was (and is) pretty much all extreme metal from Slayer on if not the melding of the rhythmic intensity of "Exciter" with the epic sensibility of "Beyond the Realms of Death" - that is, the melding of the 'punk' influence and the classical and folk influences? Hell, go check out some of the classic Slayer interviews of the early and mid 1980s, When asked about their biggest influences, they rattle off the usual metal suspects - Iron Maiden, Judas Priest and Venom - and D.R.I., Dicharge and The Exploited, and Bach and Beethoven. I don't know if it gets much clearer than that.