In TLS 1.3 all suites (such as TLS_AES_128_GCM_SHA256) have forward secrecy so it isn't even explicitly called out.
In these modern modes (and in other modern protocols like SSH) the two peers agree random ephemeral keys (these days with Elliptic Curve Diffie Hellman) and long term private keys are only used to sign things to prove who you're talking to over the resulting securely encrypted connection.
So if you break RSA you can forge those signatures but you can't decrypt messages sent to and from the legitimate owner of the keys, those were, as your parent explained, secured with AES and not RSA. You would need to perform a live active attack, a MitM to interpose between the real server and its clients so as to decrypt all messages in transit.
Forward secrecy only protects against the exposure of private key material. It does not protect against broken cryptography as it depends on the cryptography to keep old messages private. That's because it works by forgetting the session keys. If you can derive those session keys again then it is of no value.
The interesting part is using a weakness in one part to help decrypt a different part.
