How do breakers prove they didn't swap a card?

By fingerprinting the card at the moment of the pull, in front of the audience. Because the scan captures features unique to that physical card, the card that arrives can be matched against the card that was pulled.

The swap allegation is corrosive because it is normally unprovable in either direction. An honest breaker cannot disprove a rumour and spends weeks defending himself. A suspicious buyer has nothing to check. A scan taken on camera ends the argument before it starts — the breaker is cleared by evidence rather than by reputation, which is a considerably better thing to be cleared by.