I think that's what would happen from Jack's point of view, but it wouldn't actually change anything in the Alpha Timeline except make Jack disappear.
I'm not sure the Juju doll works on the same principles as normal Homestuck time travel. So far, every time anyone has used it, the narrative has followed them, and whatever world people end up in with regards to the pins stuck in the doll remained the Alpha Timeline; otherwise we wouldn't have seen all of the other Felt henchmen come back when Spades Slick pulled their pins out.
There's no reason to think that pulling
Snowman's pin would work differently (except that the change might be far more drastic, and admittedly take him to a timeline that could never qualify for Alpha.)
Although actually, it's not really clear how it works right now, since pulling their pins seemed to simply bring them back to life without changing the timeline in any other way. Still, for whatever reason Slick was unwilling to toy with
Snowman's pin -- possibly because he wasn't sure he was up to seeing her again, possibly because of his innate hatred for Weird Time Shit.
Still, it'd be an amusing Chekov's Gun if pulling the black pin effectively retroactively unhappened the Green Sun and depowered everyone who relies on it.