I could see states implementing a requirement (maybe a handful of states) that required the birth certificate to be shown to the Attorney General in each state to certify who they are. But even if you disqualified the President in a couple of states....write-in ballots would be used by most folks to still force the issue into the next round.
Write-in ballots are tricky though for those who utilize them and not all judges are like the ones in Alaska. Also, we have a Supreme Court that is generally fair.