Let's imagine that everything you say is 100% correct. LLMs can't access materials for developing weapons without weapons tests conducted at their behest, and humans choose not to - of their own volition - use world-ending weapons.
Let's also imagine for the purpose of conversation that one of the things that can be gotten at an extremely high skill level through increased intelligence is the power of persuasion.
A "superintelligence" persuades humans to perform weapons tests and then to give it the ability to use the weapon.
I think everything you've said is reasonable except the bit where you think step one of an "AI" exterminating humanity wouldn't look like a human doing it. In my opinion, the most likely vector for disaster is a human being who is "guided" by a computer model to do something unfortunate, not the model doing so directly.