Were it not for them saying they would be in town to sign the lease and accept payment, I would agree with you. If not for that, I would call this an unusual arrangement, but not 100% a scam.
My rule though is to never put down money down, except for a reasonable application fee, on a place I can’t physically see the inside of, in the company of someone with a key. That would have stopped me from going ahead with the deal altogether, and it stops 99% of all rental scams.
This is a great rule but there are rental scams where they rent the place to show you, eventually give you a copy of the key and the place was never for rent.
Check broker's credentials, do property records searches, be diligent, particularly when dealing with an independent owner.
My rule though is to never put down money down, except for a reasonable application fee, on a place I can’t physically see the inside of, in the company of someone with a key. That would have stopped me from going ahead with the deal altogether, and it stops 99% of all rental scams.