Did George W. Bush show Congress a long-form birth certificate? Did Bill Clinton? Did Ronald Reagan?
This is where your argument falls apart. The Constitution says a President must "qualify," but doesn't say how he does that. The courts have therefore ruled that a president-elect has "qualified" if Congress accepts the electoral votes.
Really. Care to cite the case?
BTW, There is no such thing as a "President Elect" until AFTER Congress accepts and ratifies the electoral college results. This means that since there is no President Elect before the acceptance of the electoral college votes that it is impossible for him/her to "qualify" for anything. In addition, since the ratification by Congress is REQUIRED before you can even have a President Elect, the passage requiring a President Elect to qualify has nothing to do with the ratification process.
Obama's own legal team admits that the process is exactly as I've described it. They used the Twentieth Amendment, section three as their defense against anyone other than Congress having the "standing" to question the eligibility of a President Elect.