Well, I'll be darn ... confirmed ... it is a 2.4.x and the Google APIs Client Library is present (vr. 1.1.7). Boy, that was a short-lived jaunt ... wasn't it!
The thing about 'exchange' ... may not be a true 'exchange' server ... in investigating the mail DNS records related there is an 'inbound message exchange' involved in the flow of mail from Moodle server ... probably back to teachers/students within that domain.
Matteo is correct ... and just to add ... troubleshooting tip ... follow the path ... moodle -> phpmailer -> Exim -> whatever the next hop is ... the inbound servers? Server is hosted in the 'cloud' (not amazon/google/azure, but a hosting provider). Is Exim setup with a smarthost? (all mail sent to Exim MTA is directed to a mail server to handle relays [the 'smarthost]' or in other terms 'an STMP relay').
Has the SMTP relay(s)/server(s) been picked up by SpamHause or other? Exim mail servers logs *might* show that.
All in all ... a convoluted mail setup.
'spirit of sharing', Ken