Asymmetric cryptography is only used in the TLS handshake. The TLS handshake ends with both parties having a shared symmetric key only they know. The handshake

Asymmetric cryptography is only used in the TLS handshake. The TLS handshake ends with both parties having a shared symmetric key only they know. The handshake can't be completed without the server having access to its private key. Everything after that only needs the symmetric key. That both sides know, and thus both sides can generate whatever they want to put into a recording, and you can't verify if it is correct or not. All a third party can verify given a detailed recording is "this client did access the bank web server and completed a handshake with it".