Try to think of how a bad actor might use what you are suggesting to steal someone’s identity. The trouble with any identity verification system is not just how inconvenient it is, but also how criminals will try to abuse the system.
Some kind of entity would have to be in charge of storing and verifying the DNA data. Once that database is created, there will always be the potential for exploitation. Also consider what happens if, or more likely when, that entity changes hands and is run by new people with new agendas? That might not work out so well for the people in that database.
The government has my fingerprints and my drivers license photo. I am not interested in sharing that with any for-profit company. I would be even more resistant to anyone wanting to collect my DNA regardless of the reason.
The answer, at least in my experience, is that schools offer classes throughout the day and it is up to the student to select classes that fit onto their schedule. Unless a class is only offered in the morning, it is entirely possible to attend classes that do not start before noon and still get a degree.