There is more information on this subject here
https://www.theamazonbasin.com/wiki/index.php/Lightning_Skill_DamageIt does not apply to a missile's own damage, so it does not apply to the nova and bolts released by the last two Claws of Thunder charges, or the piercing lightning bolts released by the second Phoenix Strike charge (fixed in D2R).
"(fixed in D2R)" in the end — so maybe it was fixed in Resurrected?
Oh shit, you appear to be right!
https://diablo2.wiki.fextralife.com/Patch+Notes#Patch2.6Quality-of-Life Updates and Bug Fixes
Gameplay
Assassin
Fixed an issue where missiles created from Martial Arts charge up skills would not gain the benefit of +% To Elemental Skill Damage modifiers.
Looks like Assassin specific skill pages on Amazon Basin need to be updated, because they don't mention this. Can't edit my original post anymore sadly.. but it seems like +% skill damage is indeed fixed and works with Phoenix Strike etc