I have to disagree with you on that. Impeachment is the prescribed remedy for the removal of a lawfully seated President; if in fact, Obama is not Constitutionally eligible for the office, he was never lawfully seated.
If I wanted to remove a headlight from the 2004 Jeep Grand Cherokee, I could follow the instructions in a manual or find a video showing me how to do so. That would be fine, provided the appropriate headlight was there in the first place. If on the other hand, somebody had jerry rigged a headlight from a 1968 VW Vanagon in its place, the manufacturer instructions for removing the Jeep headlight from the Jeep headlight socket would largely be irrelevant; something altogether different would be required.
While the constitution does not directly address the possibility of a fraudulent elevation to the presidency, the fact remains that there is no prescription for removing a president or pseudo-president from office outside of impeachment.
There is no constitutional authority for voiding and election nor declaring the president (after assuming office) to be illegitimate.
Your car analogy is not quite appropriate. The constitution is not a detailed description of government but merely the basis of the laws, a framework. For your example to work one would have to find an actual federal law for removing an illegitimate president through means other than the impeachment process.