How to send local mail via postfix to another server

TLS just enables encryption on the smtp session and doesn't directly affect whether or not Postfix will be allowed to relay a message.

The relaying denied message occurs because the smtpd_recipient_restrictions rules was not matched. One of those conditions must be fulfilled to allow the message to go through:

smtpd_recipient_restrictions =
    permit_sasl_authenticated
    check_recipient_access hash:/etc/postfix/filtered_domains
    permit_mynetworks
    reject_unauth_destination

To explain those rules:

permit_sasl_authenticated

permits authenticated senders through SASL. This will be necessary to authenticate users outside of your network which are normally blocked.

check_recipient_access

This will cause postfix to look in /etc/postfix/filtered_domains for rules based on the recipient address. (Judging by the file name on the file name, it is probably just blocking specific domains... Check to see if gmail.com is listed in there?)

permit_mynetworks

This will permit hosts by IP address that match IP ranges specified in $mynetworks. In the main.cf you posted, $mynetworks was set to 127.0.0.1, so it will only relay emails generated by the server itself.

Based on that configuration, your mail client will need to use SMTP Authentication before being allowed to relay messages. I'm not sure what database SASL is using. That is specified in /usr/lib/sasl2/smtpd.conf Presumably it also uses the same database as your virtual mailboxes, so you should be able enable SMTP authentication in your mail client and be all set.

Can you telnet to port 25 of any of the Google MX servers (or any smtp server at all)? This will help pin down if it's a network issue or a mail system issue.

For me, I get this valid response:

user@host:~$ telnet gmail-smtp-in.l.google.com 25
Trying 209.85.219.43...
Connected to gmail-smtp-in.l.google.com.
Escape character is '^]'.
220 mx.google.com ESMTP 3si95666ewy.62

If you can't get a connection anywhere I'd suggest checking local firewall rules.

CORRECTION: The fact you mention it works for other address would suggest it's not a local firewall restriction. Often if a mail server believes you to be a spamming it will just drop the connection before giving you a chance to send any data, so I'd still try the telnet test.