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Yeah, the hangman game does seem a bit out of place. Especially since he already knew you can't beat the curse that way.
I do have thoughts on a couple of your other questions though:
but if the only danger is in remembering specific people, what would be the harm in describing what happened and the nature of the curse, just without identifying the people?
Trouble is, since the curse wipes memories, it's tricky to figure out who did and did not know the last victim well enough to be at risk of having a memory jogged... and in at least a couple of the cases, it doesn't seem to take all that much to jog the fatal memory. Richard might have been worried that a random offhand remark in his report could spark a buried recollection and thus start a new chain of deaths.
But the fact that he leaves the audio transcriptions with everyone's names mentioned throughout leaves a real danger.
I think it's specifically remembering the *most recent victim* that triggers the curse, so it's only people who knew Richard specifically who would be in danger. Which brings me to...
What part did he play, exactly? If no one ends up discovering his work, as seemed to be his expectation, how is it any different than if he had done nothing at all?
I think the implication there is, since Richard was such a recluse, he's happy that no one knew him well enough to remember him, and thus the chain of deaths will end with him (and I think he realizes that's why he was picked for the job).
in the case of every other victim, no memory of them or their actions remained whatsoever. But in his case, the fact that everyone still remembers that there was a man hired to work on the case doesn't fit the pattern. Only his name seems to have been forgotten. Is it because they had records left behind of his involvement that they accepted to be true despite not remembering them, or did he actually avoid the curse somehow after all, or maybe just partially?
I think it's the first option. Whoever is running this operation that we work for, they knew to make a list of people who aren't allowed to see the computer's contents, and if they knew enough to do that, they've probably also figured out that written records are more reliable than memories in this case.