Worth noting that "The Handshake" episode [0] covers the key exchange using RSA. This has the downside that it doesn't support forward secrecy, meaning if an attacker ever compromises the server's private key they can retroactively decrypt traffic they previously captured.
It's more common these days to use an ECDHE exchange in which the client and server exchange keys that are generated just for this session (or at least, they should be [1]) and use those to generate the "shared secret".
In fact, in TLS 1.3 ECDHE is the only key exchange mechanism. [2]
The server then uses its long term keypair corresponding to the certificate to sign all the handshake messages that were seen previously [3].
[0] https://howhttps.works/the-handshake/
[1] https://raccoon-attack.com/