If you are blocked from your own server by Fail2Ban, you should check whether the Gmail app is running on the phone and whether IMAP/POP3 accounts are also configured there, which are collected. There may be old auth data entered and that locks you out.
You may have entered this account in the web interface, the web version does not give an error message etc… And Important, with the web version, Gmail servers directly connecting to the mail server.
The app takes over the data you set in the web interface, but the mail collections happen by the client from the phone to the mail server. So the Gmail App on your phone tries to collect the Emails with the wrong auth data and also give zero error messages – means you have no chance to notice that directly on your phone. The only thing you will notice is that your IP from your Network you are in gets constantly banned.
I had this dance for more than one year. I had installed a VPN just to unban my IP. Super annoying. The only thing I always noticed were in the Maillog that dovecot was throwing out warnings that my IP tried to auth with wrong credentials. I removed every trace of a local email client besides Gmail because I was just thinking – this was just connecting to the Google servers and this was working flawlessly. Yeah, kind of …