It's possible this might have a significant (not small, not necessarily large) impact on the smaller subset of delinquent parents that might currently have a larger double digit percentage (30% say) skipping to Canada, Mexico, or elsewhere to avoid being chased down.
Or not.
The main point here is that it's not the entire population of regular US citizens that should be looked at here, more the specific behaviour of the subset in question.
The constitution (article 11) of Mexico provides an explicit right of every person to enter Mexico without a passport (that doesn't mean every person in general -- you can be barred -- but not because you don't have a passport). You can witness this at land border crossings -- I routinely watch them let in foreigners without them (they're not following whatever uncited nonsense you read at the consulate wrongly claiming a passport is 'required' without citing any law). It is subject to immigration enforcement, but they're legally barred from requiring a passport. It won't do dick to stop delinquent people from leaving and anywhere that actually thoroughly checks passports also is a member of international child support enforcement treaties.