by Jamie Kramer.
I was going to say something along the same thing but you pretty much encompassed what I would say and then some. I didn't know about the mailtest plugin, thanks! I have been thinking about developing such a thing but hadn't gone searching for this.
As was stated already, it may be that the student sub-domain does not have SPF, DKIM, and DMARC set up such that the email server can "trust" the email being sent. So I would agree that is a natural starting point in this process.
If you happen to use the email test plugin, and you get a bunch of debug output, that output might give you an exact reason why the email is getting rejected.