If school records have been destroyed, besides the school district, where else may one find those records?

In all honesty... probably nowhere.  Assuming public school in the United States, the records would all be kept by the individual school district, and no federal or institution would have a copy. 

Thank you for your reply.

This was a catholic school, so not sure they are part of a school district, as I'm not sure how that works.

Perhaps I'll call the school district to find out.  The woman that I spoke with did not advise me to try the school district.

For Catholic schools, you might want to check with both the diocesan and the archdiocesan archives, as well as the archives of any specific religious orders that ran the school.  

For Catholic schools, you might want to check with both the diocesan and the archdiocesan archives, as well as the archives of any specific religious orders that ran the school.