After looking at some reference material I realized I had forgot one other, very important, reason that the interstage didn't separate with the S-IC. The interstage contained 4 ullage rockets that were used to help settle the propellants in the S-II tanks before ignition. These rockets provided brief, positive acceleration to the stack to ensure that there was propellant at the intake ports of the turbopumps. Without this settling there was a possibility that the turbopumps would cavitate and possibly destroy themselves.

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"A mystic is a person who is puzzled before the obvious but who understands the nonexistent." -- Elbert Hubbard