We are saying the same thing. Of course they have my signature on file. I gave it to them to be a registered voter. The issue is that polls do not check that I am who the name I give them says I am.
As you know the ballot in the box is official. There is no way to tell if a ballot in the box relates to any specific voter except that it came from Precinct 27 or 35. That's all they can tell. The check needs to happen before the receipt of the ballot.