I definitely agree that there's a gray area in regard to foreign born with foreign parentage. Ted Cruz is another example. Had he been elected president, undoubtedly, he would've faced challenges to his eligibility, challenges that had more merit and would've been taken more seriously than any faced by Obama.
Here's the rub. Who is on the hook to enforce eligibility? Is it state election officials? The Electoral College? SCOTUS? Or the voters themselves?
I also agree that the "born in Kenya" narrative was totally absurd. Why would've Obama's mother fly by herself, as a teenager, from modern, safe Hawaii to deliver her baby in a mud hut in Kenya, and then enter into a conspiracy to make it look as if he was born in Honolulu? Just completely ridiculous.
That's an excellent question, the answer to which, I think, is unclear. Article II of the Constitution specifies that only a "natural born citizen" is "eligible to" the "Office of President." On its face, this provision would not stop somebody who wasn't "eligible to" the "Office" -- for instance, somebody who was only 27 years old, say -- from running for President. But it could well be that each state (id perhaps not every state(?)) has its own requirements that only someone constitutionally "eligible to the Office" can be put on the ballot. As to that, I just don't know.
But, even there, what recourse, constitutionally, would there be if the Electors were to vote for, and elect, someone who was subsequently determined not to be "eligible"? My working assumption, and that's all it is, an assumption, is that the action of the Electors would be unreviewable. Said person, albeit otherwise "ineligible," would be the President. Impeachment and removal would be the only option to remove that person from office.