If an academic institution is managing records completed by individual investigators, should the Responsible Party be the PI or S-I? In the past, the institution’s administrators marked the university as the responsible party?

Some institutions function in the manner you describe (ie, University as Responsible Party) regarding who retains responsibility for registering the trial and reporting the results and it seems to work well in some cases. However, if your institution is conducting a large number of trials, this method may be overwhelming to the person managing notifications from ClinicalTrials.gov about individual clinical trials.