The person is not sent to prison, they are dismissed from the university.

The justice system has no place in this part of the discussion. The justice system is solely concerned with criminal allegations with criminal punishments. That means jail or prison time, or fines. Does the justice system get involved when you get fired from a job? Of course not. But that could be equally life-ruining as being expelled from a university.

The idea that this happens with no other crime is misinformed. If I am a student and I complain that my classmate plagiarized their entire paper from me, they may fail or if this is a chronic offense they may be expelled. If I am a student and I complain that a student punched me in the mouth repeatedly and then stole my wallet, the university would have it in their rights to discipline that student within their power, be it suspension or expulsion.

In a free and democratic society, as you described it, it is the right of a university to contract with a student (within some bounds) as they see fit. If at a later time they find that a student has done something that violates that contract, they have the right to find they no longer wish to do business with them.

You are correct, in the case of a prosecution, the accused has certain rights to a trial by jury etc. But you'll have to show me the private tribunal where an ADA prosecuted a college student and that kid ended up in prison without a trial by jury. Then we can start talking about the rights you mentioned.

(For the record, I am against these tribunals, but for reasons completely different than you stated. I think your reasoning is shaky at best.)