Perhaps, but it turns out they did not. If they had cared, they would have used different language, as the common law definition of "natural born citizen" at the time simply meant (and continues to mean) citizen at birth, which includes people born on US soil to alien parents living under US jurisdiction in amity with the US government.
Are you not concerned about the loyalties of a person who has a non-citizen parent?
No. Some of the most patriotic people I have known did not have US citizen parents.
Obama's father was a non-citizen. When Obama was removed from the U.S., was he still subject to the jurisdiction of the U.S. or could that jurisdiction have been questioned by other nations, such as Kenya or Great Britain?
Obama, like all US citizen who travel abroad (save diplomats), fell under foreign jurisdiction when he traveled and lived abroad as a child.
The "natural born citizen" clause recognized that simply being a citizen is not sufficient to qualify one for the Presidency.
I agree. It restricts the presidency to people who were citizens from birth, thereby excluding naturalized citizens.
There are only two kinds of citizen: naturalized and natural born. Birthers want to create a third category that never existed.
I looked into the matter and apparently you are correct. English common law seems to be consistent with what you describe; that is, naturalized or natural born.