"Would you agree this was their intent, incorporated as much as possible in the official documents of the time?"
No, I wouldn't. There's no mention of documentation requirements, let alone "incontrovertably confirmed" ones. Nothing about "familial origins and roots". You can't just make this stuff up.
Regarding the "no hint of loyalty to another nation", I've already covered that. You can't make a law about what someone feels inside. That's what you are expecting.
The founders didn't have such an expectation. Their intent, *motivated* by their desire to limit divided loyalty, was to create the reasonable rule that the President be born a citizen. In effect, barring immigrants who come here from another country and become naturalized.
Thinking about it, it might be helpful to focus on the two words, "motivation" and "intent". They are different things.
I see a lot of posts about what the founders thought, what goal they were aiming for, etc. This is "motivation". It is the impetus for their action.
"Intent" means, what did they believe their words meant?
"Motivation" is the reason they chose to do something. "Intent" is the thing they chose to do. In this case we are talking about some very abstract "motivation". Guarding against divided loyalties. But those abstractions have to be translated into a real action. You can't write a law against "divided loyalties". So they translated that motiviation into the "intent" of barring immigrants from the Presidency, as expressed by the words "natural born citizen".
You may believe their effort was inadequate. You are entitled to that opinion. But it's a mistake to confuse their "intent" behind the written words of the constitution, with their "motiviation" for wanting to write something in there.