One reason to require students to read their university email is simple database convenience: if a staff member is emailing a particular student about something, or emailing the members of a particular class (etc), and if the email is delivered to a student's canonical inbox, then we can effectively presume that that email was received by the student. And we can declare ‘we presume you read
firstname.lastname@example.org’. If a student doesn't hear that a class was rearranged or cancelled, because they've decided not to read that email, then that's their problem.
Maintaining a database of students' preferred addresses doesn't work: how does that get updated?, who's allowed to query it?, what if some systems do query it and some don't?, how do you persuade (the admins of) the weird antique IT systems in Finance and in the Library to do that?
It would of course be highly desirable to let students configure that address to have it forwarded to some other preferred address. We don't have to have them use the (probably hideous) institutional email interface.
One reason to require students to send email from their institutional account is, as others have suggested, to add a little bit of security (I am more confident that a query from
email@example.com is really from Fred Bloggs than I would be about
firstname.lastname@example.org), and to keep things formal. If I were emailing
fluffybunny I would be to some extent within their private life, and if I were mad enough to be doing this, as staff, from a private address of my own (not a good idea!), then there's all sorts of way such an off-grid communication could be perceived as creating, or potentially enabling, a wide variety of improprieties.
I also am curious what the ‘personal issues’ are, that are quoted in the original question. Like others, I'm a little sceptical that this might in fact translate to ‘I'm too idle to work out how to configure two addresses in my mail client, which I am going to have to do at some point in my professional life’.
Edited to add the following (since there may be an element of talking past each other, in the answers and comments here):
Looking at the other answers here, I get the impression that those (including me) saying ‘I don't understand, what's the problem?’ appear generally to be presuming that anyone reading or sending email is doing so via a mail client, where adding multiple addresses/identities, to receive and send email, is a core function; but those saying ‘it's really annoying having two addresses, or having to wrangle two UIs’ seem to be presuming that one reads email via a webmail interface. In the latter case, I can see why having two addresses would potentially be inconvenient.
I don't forward my work or personal email addresses (plural in both cases) anywhere – I simply read them all in the one mail client. Each of these has a webmail interface that I could use in emergency, but doing so routinely would drive me insane. But someone who sees webmail as usable, through familiarity, would presumably set quite a high premium on only supporting a single address/identity, and would have a quite different take on this question and the various answers.