Sounds like the opinion of someone who has only heard two punk bands. I can only assume you have yet to hear:
The 4-Skins - The Good, The Bad & The 4-Skins
The Exploited - Troops of Tomorrow
Discharge - Hear Nothing See Nothing Say Nothing
Skrewdriver - All Skrewed Up
Motörhead - Motörhead
The Misfits - Walk Among Us
Black Flag - Damaged
Or virtually any early punk(ish) music that has a hell of a lot more to do with 1980s metal than any classical. Someone who knows more about punk (I'm pretty new to it myself) could probably list a lot more.
Since none of us can truly know what the bands were influenced by, I just tried to go on the what the bands actually said they were influenced by . And while none particularly cited a classical composer, none were citing punk bands either because the music was more or less developing during the time most of the bands themselves were. In other words, metal bands that were greatly influenced by metal bands that were influenced by classical, blues and jazz, were greatly influenced by classical, blues, and jazz. And to a lesser degree, influenced by punk.
There are a myriad of hardcore and a few black metal bands you can saddle with this 'Largely influenced by Punk' title, just stay away from my Metal bands. They may have been influenced by punk a bit, but you can't start saying 'punk's influence was the greatest ever' when these bands do not sound or only very slightly sound like they have a bit of influence from punk.
A lot of people I know used to listen to the bands you list, and I am familiar enough with them to say that they may have shared some of the same energy, and some (in most cases small) ideas, but what heavy metal got from punk, if anything, was already done before punk came around, and done quite better by more influential sources.
If if we just go ahead and say that punk influenced metal to play faster, ( Just ignore classical, jazz, and prog rock ) we are still left with song structures that have varying speeds and tempos, acoustic parts, instrumentals, solos, and a whole lot of other stuff that punk doesn't have . ( the punk was around to influence the bands in question)
I'll give you Danzig... and maybe Sodom. That's about it.