The docs specifically mention that TESTING can be used to store messages in the outbox instead of sending them directly. This does not work for me, however, as the g.outbox functionality seems not present. Many things seem undocumented, for example options like MAIL_FAIL_SILENTLY or the email_dispatched signal.
I have not found a way described in the docs that allows me to test sending emails without actually sending them. Please update the documentation to reflect the current state of Flask-Mail, if the extension is still maintained.