Wow- I never considered initiation anywhere other than where my campus would hold it. We have a non-sectarian chapel (Galbreath Chapel) on campus that holds around 200 people. It even has a basement lounge area where people could wait or change clothes if needed. I'm pretty sure all the orgs on campus used it for initiation since it was so pretty and set up perfectly. The chapel had a sign that you would put on the door that it was in use, and I don't ever remember someone trying to get in.
Even the honor society I was in used it for initiation.