I absolutely agree with you that that's a better way to think about it. But everything I learned and think I know about it says that wasn't how it was thought about until recent decades and thanks to actual legislation.In my mind, “natural born” means that you were automatically a citizen at birth because you satisfied at least one of these conditions: 1) you were born on American soil or 2) at least one of your biological parents was an American citizen at the time of your birth.
I do have to say that it's puzzling to me how the nonsense about Obama's birth certificate became such a big deal for the GOP, if everybody knew all you needed was 1 American parent. I mean I know some were pushing back against Trump and the others pushing the narrative that Obama didn't qualify. But nobody ever questioned his parentage, so the birth certificate should never have been an issue. Even if he had been born in Kenya, he was eligible under US law.