WARNING: YOU DID IT COMPLETELY WRONG.
Your change to Postfix main.cf will make your server an open relay, try to verify it with mxtoolbox.com again please.
The problem is, you didn't find the root cause. If you send an email from Gmail/Hotmail/Yahoo!, what does the returned email say? It usually provides some technical details about mail delivery failure. So, paste us this returned email so that we can help you troubleshoot and fix it.
