I support this.
To me, the salient thing about whatever word replaces lynch is not only that it conveys elimination from the game, but that it is a distinct kind of elmination from the game as opposed to nightkills (which we often shorthand to kills). It needs to be a non-cumbersome verb that conveys the sense of "sentenced to death by majority vote". It's actually not important to me that the flavor be an extrajudicial killing because I've never thought of the town weapon as extrajudicial. The town is the jury, they decide democratically who to eliminate from the game.
I think something like convict/execute/exile would be fine. I worry about "eliminate" because it's not particularly distinct from the language we would use to talk about the nightkill.
I also feel bad for Jimmmmm who is just launching a site and has presumably paid for the domain name, etc. But I do think he needs to have this conversation too and should do it sooner rather than later, especially if he's hoping to build a community of his own on his site. I'd be happy to participate in a thread over there to help Jim brainstorm new names.