…If due to the Shut & Run card the TimeBreaker moves to the Time Repair HQ, and at the same time the player does as well because their gate tile just crashed…

A: No. Remember, Shut & Run, like all Breaker cards, represents the TimeBreaker breaking free and taking an action on their own. So this is just a form of the situation in which the TimeBreaker is turning themself in. The Agent arrives back at HQ just in time to watch as TimeBreaker begins confessing and begging for mercy.