The question we have to ask ourselves is "How far should religious accommodation go?"
The state of Nebraska argues that these tests are one of the most cost-effective preventative/treatment measures that are available to them because the test can detect many problems before they become very serious. And if we allow parents to refuse the test, do we then allow them to later deplete the public coffers because the problems that could have been prevented are now life-threatening?
It is beyond cruel to allow the innocent children to suffer, and so we treat them out of the public funds, of course. But how else do we support reason and public safety if we cannot require such tests over religious objections?
Naturally, I reject religious objections as being fundamentally irrational, and against the public good. But as a matter of Constitutional law, the matter is less clear. How do we balance both what is best for the public good with the need to permit free exercise of religion? And perhaps the issue that needs to be reexamined is how much control parents really should have over their children? Which is a different philosophical question altogether, and which opens up a whole new can of worms.