As much as I like escape rooms, I'd definitely reject the idea of having a first date with a man be somewhere where I'm locked in with him for at least x amount of time and I'm being judged on my actions. All I would be thinking about is "get me out of here!"
I know there are people monitoring the situation but I still would never agree to be so dependent on strangers on a first date.
Maybe go to a board game cafe instead and play some skill based games?