[upofadown]

An attack on the PAKE would involve the attacker seeing the secret value as it was transferred to the recipient and then beating the recipient to the handshake. So there is security value in being prompt in putting in the secret value at the receiving end.

That is as opposed to sending a public key or key fingerprint. In that case there would be little value to the attacker in seeing the transfer. They would have to MITM the transfer of the key itself. If you wanted to prevent the attacker from sending bogus files you would also have to transfer some sort of signing key.

So a short, time limited, secret vs a longer public value.