Very interesting to hear what methods the tower used to try to verify his position. I wonder if it would've been worth asking if he was familiar with NDB's, because there is one just north of the field.
Since he wasn't able to set this transponder, perhaps he was bluffing about the ident, too.
Unfortunately, there's a block that occurs where the pilot says he doesn't see the beacon/tower, so the controller starts painting more of a picture...during which time the pilot transmits again that he has it in sight (and presumably starts heading towards it).
Hint for pilots, if you release the PTT key and the controller is in mid-sentence, it's likely that he didn't hear your last transmission.
My fingers are crossed that this is a student pilot who didn't know how to operate the transponder. If it's a licensed pilot, then he slipped through the net during training. Even if you don't regularly get flight following, there is still training associated with the emergency squawk code during PPL training, as well as basic operation of most equipment in the airplane.
I give the tower a lot of credit for staying calm, he did a good job.