They can't. That's the point.
The man can argue simply that he ordered a vanilla smoothie and they gave him the wrong one. When his son drank it he had an allergic reaction because apparently it had peanuts in it.
He has already won that argument because he has live audio and video of everybody in the store telling him they don't know who made the drink and what drink it was to begin with.
They'll settle. He's not going to get millions or anything, but the entire thing will give them a black eye if it ever goes to court. I mean, seriously: He can prove his son had a reaction to what they served him. They don't even have the first idea what it was that was served or who made it. That, in and of itself, is negligent.
Think about it. Even in a non-eventful transaction, surely out of 4 or 5 people at least ONE could remember the guy being there and what he ordered only an hour and a half before.