The secondary air (SAI) pump on an E36 injects air into the exhaust for the first minute or two after a cold start to help the catalyst light off faster — purely an emissions/warm-up device that does nothing for running performance once the engine is warm.
So "does the M3 (or any E36) need it?" — mechanically no, the engine runs fine without it; emissions-wise it's part of the system and deleting it sets a code.
When it fails (the pump
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