Failed to send email. Reason: Invalid login: 535 Authentication failed

[jannis]

Try setting “secure” to “false”.

The “false” is confusing – it’s still secure. Have a look here, for example:

Since there's a builtin Mailgun integration, do I need to update the code?

This likely needs to be "secure": false. The naming is confusing, it doesn’t mean the connection isn’t secure, it just means the initial connection is opened without SSL and then STARTTLS is used so your server and Mailgun’s servers can agree on a compatible cipher to open the secure connection.