Is "Why would a given thing require/not require a notary" on topic?
Yes. That type of questions often traces back to concepts such as legislative intent, and legislative intent is in scope on LawSE. Even if the requirement of a notarized document in a specific scenario is not premised on statutory law, the procedural and evidentiary implications of notarizing a document suggest that this type of questions should not be considered off-topic.
It just seems that such on topic questions would not elicit answers other than the broad, general reason for requiring a document to be notarized; namely, for the notary to witness/certify that the person is signing the document out of his free will rather than the signature being coerced or forged.
would there possibly be some general legal principle that governs when forms do and do not require a notary that could provide a closed form answer?
No. If one legal principle were general enough to determine this issue for any context, that principle would be broadly incorporated in the legislation. That would obviate the need for legislators and contract parties to specify the circumstances where notarization is necessary. The reality is that statutory law and contracts prescribe the particular scenarios that require notarization.
There are matters for which most jurisdictions (if not all) acknowledge only a document that is notarized. Examples can be found in real estate, mortgages, and procedures for being authorized to exercising certain professions. But that element is something the policymaker decides for each kind of scenario.
