@something_amusing
[ignoring everything in this thread that you didn't post]
You are getting two errors.
The first says Unable to email via SMTP:smtpserveraddress:587 [email@domain.com] Failed to add recipient: @localhost [SMTP: Invalid response code received from server (code: 501, response: 5.1.3 Invalid address)]
The second says Unable to email via php mail function: mail() returned failure
osTicket tries to send an email via SMTP: from email@domain.com to @localhost. Since @localhost is not a valid email address it fails completely.
Since it failed it attempts to fail back to phpmail to send the email. Which once again fails probably for the same reason.
I'm curious why your User has an email address of @localhost instead of well an email address.