I expect it depends on how it's treated. Jay and Acacia ended up taking out one or two Mary Sues that were planning to die (permanently) of their own accord. The Mary Sue prototype specifically parodied in the A Trekkie's Tale did, too. So it can certainly be done badly enough to be a Mary Sue trait on its own.

It's not always such. A character like Riala Goldentail, from Vengeance Quest, is pretty much defined by her drive to achieve one specific goal, and her willingness to drop everything to achieve that goal. If she abandoned her post and got killed as a result, it'd be stupid, but it'd be stupid that fit exactly within her character and goals. And even if the story ended on that note, readers would expect her actions there to be criticized just as other mistakes she made were.

On the other hand, it's far more common for a Sue to end up with the entire cast pining over her for eternity and some opportunity for wangst.

is that, according to The Original Series, that means her 'legacy' will live on and continue to affect the canon, which means that she has to be killed before the death that she plans.

As for 'normal' Sues, there are many different types, and this one sounds just as Sueish as most others. Go ahead and kill her.