How can I configure Postfix to send all email through my Gmail account?

You need to enable TLS in Postfix's SMTP client, since Google requires it. This is indicated by them in the message "Must issue a STARTTLS command".

In /etc/postfix/main.cf, you want something like this:

smtp_tls_policy_maps = hash:/etc/postfix/tls_policy

and then in /etc/postfix/tls_policy:

[smtp.gmail.com]:587 encrypt

The left hand side in tls_policy must appear exactly the same as your relayhost entry in main.cf.

Remember to run postmap on /etc/postfix/tls_policy after creating or changing it as required.

You can find more details in Postfix's TLS documentation.