I had my cards off, and did a lot of work in the doors, lubing sliders, replacing a speaker and a door pull, replacing the rubber gaskets under the handles, replacing the foam shields over the inside handles, and replacing the plastic card sheets with foam panels. And yet, I never noticed that anti-theft plate or its rubber cushion. It was only later that I heard it could cause a rattle with the window, and it is the only thing I have heard of that rattles with the window. But I had no rattles to bother about, and never looked at that plate.
So yes, do look into the door, with an awareness of that plate as a potential rattle-causer.