I agree with the above comments. I'll also add that the more history a school has with the program, the stronger likelihood there will be more investors, mentors, ex-searchers, etc. at your fingertips. Unpopular opinion, but people like talking to those they're connected with so you have a better shot at getting time with someone if they are an alumni of or connected to your school.
All that said, I will say the SF community is very open and helpful so if this is truly your goal you can make it work wherever you go - just get involved and get out there!

Fundraising times are a good proxy for this. Someone from HBS or Stanford can sometimes get their searchfund raised within a month. If you go one tier down to Chicago or Columbia, it will probably take 2-3 months and it falls from there. Unfortunately, where you go to school is by far the greatest predictor of fundraising success.