Not sure this is the real reason or not, this is what Bredesen says:
Gov. Bredesen has reservations about a proposed federal requirement that would force states to screen driver’s license applicants more closely.
. . .
Bredesen said the proposal is an unfunded mandate.
He said national security issues don’t belong in the hands of driver’s license bureaus of the states.
He’s correct in pointing out the federal nature of the problem. This appears to be another feel-good, look-at-us-getting-tough measure by Congress. If it was really serious about stopping illegal immigration, it would tighten up the border and go after employers. But that would alienate too many business folk. So apparently we are left with symbolic legislation.