Given that this other half you speak of is entirely theoretical, even to the point of being theological, doesn't this compromise your entire notion of what makes a Folk Song a Folk Song?

It's about as theological as my belief in evolution - which I also can't observe in practice. It's the theory that best fits the evidence.

absolutes that account for every single variation of every single song that has come down to us

I think the combination of individual variation and differential survival does, in fact, account for every single variation of every single song that has come down to us - not because they're theoretical 'absolutes' but because I don't know what other factors there are.