Even someone supposedly ineligible to be president has the right to due process to determine whether this or is not the case.
What person or body do you propose make this determination? The Constitution provides no guidance. It provides no guidance as to dealing with someone elected who is ineligible.
Do we really want to turn such determination over to an already overly powerful court system?
To my mind it is clear the Founders intended the determination of eligibility to be left up to the members of the Electoral College. It provides no remedy for an error on their part.
I agree, but IIRC the courts declared that even members of the Electoral College do not have the "standing" to force the issue.