Turn that around. What do you think *should* happen should a person be found not eligible to the Office after having taken it.
I know what usually happens when it's found out that a job applicant lies on his application or provides false information as to ones educational background and experience. She gets fired, and often the company security guards will escort her off the property, and rather quickly.
How to you fire a person unlawfully occupying the office of President. Are they really President if they were never eligible in the first place? If they aren't then they cannot be impeached, since that's for removing a President, which they would never have actually been per the Constitution's requirements.
Suppose they were actually only 34 years old, or had only lived in the country for 12 or 13 years. The Natural born citizen requirement is not the only one in Article II.
You asked — “Turn that around. What do you think *should* happen should a person be found not eligible to the Office after having taken it.”
I did state that already. I said Impeachment (which I said I had always considered the proper procedure). I was rooting for the conviction of Clinton during his impeachment... :-)
It shoulda been a slam-dunk...