They are considered by American law, in all respects, to be "natural born citizens" of the United States, having been born in an American hospital on American soil.
If Barry Soetoro was born on American soil, I would willingly afford him, in accordance with the law, the responsibilities and rewards of citizenship.
As would most Americans. However, THAT'S the part he is unwilling to conclusively prove beyond a doubr.
It is wise that Vladimir Putin, for example, can not foster a child while vacationing in Hawaii and then use the wealth of the Soviet Union to acquire the office of the president for the child at a later date.
This is where you lose me in your argument. I don't understand the point you are trying to make. Naturalized citizens are those who were born else where and renounced their citizenship in the country of their birth to become American citizens. If Putin were to adopt a natural born American child, there is nothing in our laws to prevent that child from becoming the President (assuming that that child retains his/her American citizenship), regardless WHO buys the office for him. I'm not sure how this even factors in the debate. Barry Soetero CLAIMS to be a natural born American citizen but, so far, has been unwilling/unable to prove it beyond a doubt. His origins are exceptionally cloudy and muddy, which adds to the controversy.
IF he were born in Hawaii, as he claims and IF his father was a naturalized British citizen, giving Barry dual citizenship, his father's nationality (natural or otherwise) would be immaterial IF Barry had renounced his British citizenship by age 18. IF Barry DID NOT renounce hsi British citizenship, then you are correct, it is a consequential issue, making him ineligible.
The controversy over what passport he traveled to Indonesia on adds fuel to this fire but, for the most part, is based on speculation. It creates an interesting tangent to the debate that takes the focus off the debate itself. The primary debate remains rooted in whether or not he is actually a natural born citizen and CAN PROVE IT.