That's why the won's fraudulent SSN matters; The Discovery process.
The courts can decide that the SSN has no bearing on his eligibility, but at least it opens him up to criminal charges, based on the evidence, upon which the Congress can then act.
Why would a Conservative be against this?
BTW, I think the Clinton case shows that the people will not support impeachment based on what they view as nit-picking. I disagreed with them, but their position, as usual, carried the day.