That could be a possibility, but right now the onus is on the candidate. We are forced to trust the states to do so much in our lives while keeping a reasonable society from falling apart. If we can count on the state to honestly review and assess a candidate's birth certificate, then we should be able to count on another state to supply an accurate certified copy of that record.
There is frequently more than two candidates for president on the ballot. Arizona for example listed five.
John McCain (R) 1,132,560 54%
Barack Obama (D) 948,648 45%
Bob Barr (Lib.) 11,506 1%
Ralph Nader (I) 10,365 0%
Cynthia McKinney (Green) 3,107 0%
I think it would be a waste of taxpayer dollars to send teams to each state to verify the original, especially if those states forbid by law any access to their records except to those that are allowed to see them. Election committees are not normally on those lists of people. Arizona law as it is currently written protects access to vital records. While they allow certified copies to various people including representatives of the person in question, it is not the same as access to the original. I was not able to find any law that allowed even the person listed as the subject of the birth on the BC to view the original.
I think the legislators in Arizona are letting down their constituents by sponsoring such a bad bill. Did they even stop to think if they themselves could meet those requirements if they ever intended to run for president?