NO NO NO!!! They did not want anyone with divided national ALLEGIANCE!!!
I already covered this. You are talking about motive and purpose. The way they chose to implement that purpose, the rule they imposed, was to prohibit foreigners from coming to America, being naturalized, and becoming President. That's the effect of the "natural born" requirement. It is totally in keeping with the motive.
That you think they *should* have been more restrictive doesn't change what they in fact did.