I have a personal theory that making first or second cousin marriage illegal would result in a great deal of complaining and likely legal challenges from certain groups that they're being discriminated against on the basis of racial/ethnic origin. Take a look at a global map of the percentage prevalence of first-cousin marriage.

I say this having discussed the phenomenon with friends who immigrated from Pakistan and all of whom have met first-cousin married couples that had one or more children with birth defects or mental disabilities that could very well have been the result of it.