There are many situtations where having the "student" print the certificate is desireable. Uh.. how many online courses did you take back in the day
In one of my training environment, we provide the certified training and other organizations Human Resource Depts provide the key to get into one of our courses 24/7. We keep a record but it is the organization that needs a copy of the certificate. The "student" gives the certificate to the person who authorized them to take the course by giving them the key. This is common.
Also consider certificates as non academic motivators. "Hey Mom, look at the cool certificate I got for learning how sensitive some old people are about technology!" Humorous example, but certificatescan be like rtibbons or stickers.
Hope this helps answer why a student should be able to print their ceritifcate.