When a Gold Gear token adds a rune, is that rune added exhausted?

To answer your question: No, a Gold gear token does not add a rune to the board.

According to the Unleashed patch notes, a Gold gear token's ability is: "Gold gives you one Power when you exhaust and kill it."

When you activate this ability, you are generating the resource Power and adding it to your Rune Pool (as per Rule 164 and Rule 416). You are not "channeling" or adding a Rune card to the battlefield. Because no Rune card is being added to the board, the question of whether it enters exhausted does not apply.

If you have any other questions about how resources or tokens function, please let me know!