The reliance on LLMs is unfortunate. I bet this mystery could gave been solved much quicker by simply looking at the packet capture in Wireshark. The Wireshark dissectors are quite mature, SSH is covered fairly well.
Seems because dumping the session keys is not at all a common thing. It's just a matter of effort though - if someone put in the time to improve the SSH story for dissectors, most of the groundwork is there.