In context, it is clear that Paine is juxtaposing the open British monarchy with the closed American executive, "open" meaning accessible to "foreigners" and "half a foreigner," where the American executive is closed to both. One cannot be in "full natural... connection with the country" if one parent is not a citizen.
-PJ
Clear as mud. Paine only says that the President is never a foreigner. Not a half-foreigner, a foreigner. And nobody disputes that. The President is a natural born citizen and not foreign born.