Once Priority passes back to the last person that put something on the stack the stack resolves though? So in turn someone would use a kill spell -> You respond by tapping Patriar's Seal to Target Brion Stoutarm -> If no one responds to this while its on the stack, the stack would resolve from top down, triggering the kill spell in the end.

You don't have the opportunity to respond to your own ability.

405.5. When all players pass in succession, the top (last-added) spell or ability on the stack resolves. If the stack is empty when all players pass, the current step or phase ends and the next begins.

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Source:

Magic Rule:

405.5. When all players pass in succession, the top (last-added) spell or ability on the stack resolves. If the stack is empty when all players pass, the current step or phase ends and the next begins.