TL;DR: All true about Venom, but I vote Slayer.
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I think the basic genealogy of exteme metal simplified to Motorhead --> Venom --> Metallica --> Slayer --> Hellhammer/Frost & Bathory
...is more or less true. I agree with the imagery of Venom being the thick branch out of which come thrash, death and black. They are the proto-extreme metal band. What Hendrix is to shredding, Venom is to Extreme Metal.
Incidentally, I wrote this about a month ago (after a tired debate, are they black, yes or no):
Is Venom a black metal band? The answer is not a simple yes or no.
Venom is a heavy metal band but they are also the first extreme metal band.
They are the fathers of thrash, death and black metal as Motorhead are the grandfathers. Venom also baptized black metal and forever connected it to satanic imagery.
Venom's child, Bathory is the first real black metal band that musically and aesthetically defines the genre as we understand it today. Hellhammer, the other contender (my vote goes to Bathory), is also a direct child of Venom ("Venom kills music, Hellhammer kills Venom") and like Bathory influenced both death and black metal bands. Both of them, like their daddy Venom, were called death metal, black metal and thrash metal, when the labels hadn't been solidified yet.
Of course Venom's influence on thrash, including of course the bigs and the firsts on both sides of the Atlantic is well established and documented. Metallica (Mustaine included) and Slayer would be the first to acknowledge that Venom is right up there with Motorhead, Priest, Sabbath and Maiden in terms of impact.
Death metal? Possessed and Mantas were worshipping Venom on their demos so hard it's almost funny. Chuck names them as the point zero in death metal as far as he was concerned and his opinion carries a lot of weight.
In conclusion, taking all of the above into account, to me it is valid to call Venom proto-thrash, proto-death and proto-black. They directly inspired and influenced ALL pioneers of those three genres.
Venom is one of the most important bands in metal history that precedes and inspires (multiple) genres.
But the main thing is they also play some damn fucking mean, filthy and enjoyable heavy metal.
Hacing said all that... Slayer is the Sabbath of extreme metal in terms of impact, cultural and musical. Just not in terms of "who did it first". Slayer didn't do it first. Nevertheless, I find them more important than Venom (even if they are essential, so is everyone else named so far) and more to the point, their musical influence still being relevant, if not dominant in the whole of extreme metal. More than Frost or Bathory. Slayer is every-where. Inescapable.
Metallica is probably just as important to extreme metal as to any other genre of metal or outside metal. But since there is no lack of recognition, I think it's proper the title goes to Slayer, since being extreme always remained a part of their DNA. But, it is very important to stress that Metallica were BY FAR the most sonically brutal thing in metal in mid 1983 and a year earlier with No Life Till Leather (extremely popular in the extreme underground). Their first tours were with speed freaks Raven (and they eclipsed them as the story goes) and Venom (who they also eclipsed the story goes). They just immediately make a statement that brutality and blind speed are not dominant in their DNA, in fact they could play the opposite if they so wished; with that album intro in 1984 and Fade to Black, although how BRUTAL is Fight Fire with Fire for fucking 1984? Until 1986 (Battery/Damage qualify just fine as some of the most brutal of the year) they are also inescapable and by sheer reach, necessarily and unavoidably extremely influential to everyone, including the extremes.
But. Slayer.