However, it took only a very quick explanation to reassure them - if its on the internet, there will be a system administrator who can see all of it - at least with a self-hosted system (as ours was) that system administrator is an employee of the university and subject to the professional standards of that institution. Needless to say, there was never any issue over confidentiality, but I was able to support them with a couple of issues around student submissions, which would not have been possible without proper admin access.
One compromise is to create yourself a separate user account for day-day use, so that the site admin account is only used when absolutely necessary - but it is necessary, and it will have access to everything, including the database.
As Just H says, if there is a trust issue within the university, you need to address that rather than anything else - its far more crucial than any technical aspect!