They did. They used the word "citizen", which was common in Switzerland, but not so much in England.
In England they used the word "subject."
That's the common law word you know. They didn't use it.
Implies they didn't use the common law on the issue of "citizen."
The Founders didn't want to use the word "subject." They couldn't say "subject" because they had just fought a revolution to get rid of the king. The word "citizen" was available and in use. That did not mean that they had to adopt Vattel's definition of "natural born citizen."
The Framers may not have been agreed on what "natural born citizen" meant. Their chief concern, I'm told, was to prevent foreign monarchs from coming to the country and getting elected president (as happened with elective monarchies in Europe). The question of whether it made a difference if your parents were naturalized a year before or a year after you were born may not have been on their minds. At a time when every lawyer had to know his Blackstone, it's likely that Blackstone's idea of what "natural born" meant would be in most lawyer's minds.
P.S. What's so special about Canada? The US is a very big country with lots of places to hide, most of them further away from Seattle (and Honolulu) than Vancouver. It's not like our Northern border is some impassible Iron Curtain.