Seems like the agent wasn't effectively closing, instead he was setting up a distracting, artificial "contest" between two prospective clients. If he truly was

Seems like the agent wasn't effectively closing, instead he was setting up a distracting, artificial "contest" between two prospective clients. If he truly was "always closing", he would have given you his attention and sold each of you individually - and hard - on the merits of the place. (And maybe, parenthetically, mentioned how much interest there was in it among other prospective renters...)