Unless they intercept the handshake as a proxy and have access to everything after that.

You’re thinking of a MitM proxy, and generally speaking what you described is not a risk when using public wifi.

There are two ways you can set up a MitM proxy:

1. Forward all traffic back to the user unencrypted (over HTTP)
2. Forward all traffic back to the user encrypted (with HTTPS)

The first option will result in prominent warnings in all modern browsers. If the sites in question implemented HSTS and the user has visited them before, the browser will outright refuse to load them.

The second option will result in even more prominent warnings that you have to go out of your way to bypass in all browsers. The only way it wouldn’t would be if one or more of the following is true:

1. your computer has already been compromised and root certificates were installed, such that the proxy owner could use it to sign the certificates
2. if a certificate authority was compromised, or
3. if the site itself was compromised (e.g., if the attacker was able to acquire the SSL cert used for the site or the credentials necessary to generate a new, trusted one).