So, if you declare, that counts as immunity and they would never investigate? Seems like a big assumption. For the record, I'm not advocating not to declare, but there are emergencies and then there are emergencies. I don't think declaring for every minute malfunction is always the best option as it can potentially open up a door that you don't need or want to open.

No, I never said or implied anything like that. I said, "If they're going to investigate whether you declare or not, then there's no reason not to declare."

Your assertion was that the "significant downside" to declaring is that "the FAA will now investigate". That is, IF you declare, THEN they will investigate. A counter example (declared but the FAA did not investigate) proves the assertion false, but isn't a claim of the opposite. If someone says "all cows are brown", pointing out a single black cow is not making a claim that all cows are black.

So far, nobody has provided any examples of investigations that happened only because they declared, that wouldn't have happened if they hadn't.

Adam is correct, they do investigate. Not saying this is a reason to not declare; I definitely would. A friend of mine lost his engine over Alpine tower along the Hudson river. He secured the engine and returned to FRG. When he contacted the tower he told them he had OEI but did not declare. Tower recorded it and rolled the trucks, FAA investigated and claimed he overflew 3 closer (better choice) airports and required a him to take a 709 ride plus a full maintenance log review of the airplane.