Given the tight control of information that has characterized O's campaign, I'm not so sure they "know," as much as they suspect, in spite of all the lockstep blather. With the possible exception of Hillary. And what she might know could be something entirely unrelated to the question of citizenship, or "natural born."
I agree with your take on this situation, and given that his supporters are now quoting the opinion of the Circuit Court judge in the (ill-conceived and executed) Berg case, to the effect that the voters decide the qualifications of the President, it's one more nail in the coffin of Constitutional authority.
And I can't help wondering if that's been O's objective all along with this issue.
I think it probably is. If this issue is allowed to slide, why not any issue? The Constitution becomes simply a general guideline which may be honored or not, depending on how convenient it is for the people in power.