Believe what you will. We can all sit here and discuss how things “should be”. Doesn’t matter what the theory is when reality is very different. One thing is certain, precedent has been set for over 200 years. And we shouldn’t expect anything different now, or in the near future.
Yes, the precedent was set, until 2008. Chester A. Arthur was the one exception, and he burned all his personal papers prior to his death to conceal the details about his father's citizenship, and his own birth.
Every other president and vice-president from Washington through GWB met the Vattel definition of "born in the country, to parents who were citizens at the time of their birth", or they were citizens at the time of the ratification of the US Constitution.