does killing a gold gear token activate the sivir battle mistress legend ability?

No. Sivir, Battle Mistress (Battle Mistress) has exactly two triggers: "When you recycle a rune, you may exhaust me to play a Gold gear token exhausted" and "When one or more enemy units die, ready me." Killing a Gold gear token satisfies neither.

A Gold token is a gear, not a unit (Rule 187.5), so its death never counts as "an enemy unit dies." And killing a gear is not recycling a rune — recycling is a specific action that moves a rune back into your rune deck, not a Kill (Rule 428). Note also that when you kill the Gold via its own ability ("Kill this, [E]: [Add] [A]"), the kill is a cost, not a recycle, so nothing about that sequence touches Sivir.

So the only ways to fire Sivir are a genuine rune recycle (for the Gold token trigger) or an enemy unit actually dying (for the ready trigger).