what would much more easily and credibly be explained by... Kiss, King Diamond, and Celtic Frost.
As u/sagathain has already said, there's nothing Norse or berserker about it. The true originator might be Arthur Brown, frontman of The Crazy World of Arthur Brown, who was wearing corpse-paint-like makeup in the 1960s:
as part of a demonic/horror stage persona (but with very non-black-metal music). Makeup like this became a part of horror rock, with Screaming Lord Sutch using it at times, and Alice Cooper bringing it to wide audiences (and Alice Cooper was influenced by Arthur Brown), and this tradition has been continued, e.g., by the Misfits.
KISS deserves at least an honourable mention due to Gene Simmons' demonic makeup (a fitting match for his stage name of Demon). However, while his bandmates also wore black-and-white makeup, they were much more solidly in the Glam tradition than horror or demonic (imagine a black metal musician in Catman makeup!).
Black metal pioneers King Diamond (Mercyful Fate) and Tom Warrior (Hellhammer and Celtic Frost) brought the look to black metal. Hellhammer, a ludicrous laughing stock as far as mainstream metal critics were concerned, was enormously influential in early Nordic black metal, with Mayhem members taking stage names from Hellhammer: Hellhammer, Messiah, Maniac, and Euronymous (from Hellhammer's "Eurynomos"). Mayhem also featured Dead (Per Ohlin), who made "corpse paint" "corpse paint" rather than just horror/demonic makeup. Dead wore makeup to look like a corpse (his stage persona), a persona adopted not so much for the stage as an expression of his obsession with death (wearing buried-and-dug-up clothes, and sniffing a dead crow before performing, etc. are not signs of normal behaviour). Mayhem then proceeded to implode in suicide, murder, and arson, but corpse paint lived on as a standard part of the black metal stage show and album cover.
Postscript: Arthur Brown has continued to wear makeup on stage into the 21st century, but is often rather more colourful than corpse-like: https://commons.wikimedia.org/wiki/File:The_Crazy_World_of_Arthur_Brown_(15559408036).jpg
I can't really answer where it came from, but to address the point of it coming from berserkir, that is beyond a doubt exactly as you describe - a sneaky "pagan" origin that is completely ahistorical.
As far as I'm aware, there are actually zero descriptions of berserkir wearing anything that looks like corpse paint. Note: this is coming from the literary descriptions, which have a highly uncertain relationship to historical reality. We don't have any detailed contemporary descriptions of berserkir, and quite scant archaeological evidence, so any accounts of what they were like necessarily comes from centuries after the fact.
However, even with that disclaimer, I don't think a single source mentions them wearing any sort of face paint. The most explicit (and hyperbolic) account comes from Snorri Sturluson's Ynglinga saga,
his men rushed forwards without armour, were as mad as dogs or wolves, bit their shields, and were strong as bears or wild bulls, and killed people at a blow, but neither fire nor iron told upon themselves. These were called Berserker.
Roderick Dale, whose Ph.D. thesis is the most recent broad study of the berserkir, similarly doesn't mention any indication of paint of any form as an attribute of the berserkir.
I hope someone else chimes in with a more positive explanation of where it did come from, but hopefully that at least confirms your instinct that it isn't because of a very old Norse tradition!
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