The reason this case is being argued is that the US Federal Government doesn't want to start giving asylum status to what aren't asylum cases. The German family is arguing that they should be granted asylum here because they will be persecuted in Germany, where persecution is defined as not being able to homeschool. Asylum status here is reserved for people who are experiencing serious threat of actual discrimination and death. So in order to secure it, the German family's lawyer has to argue that this family is having a fundamental right violated by not being able to homeschool their child in Germany.
If the US agrees to this, then we'll have to start giving asylums out to all kinds of people. Muslims from France can say that having the veil banned there means they should be able to get asylum here, for example, which is extreme. They will be put in the same category with, and potentially take spots away from people who are actually facing the threat of death and much more serious violations.
That's the real lens through which you should be looking at this case. The government isn't saying that banning homeschooling would be totally fine. Yes, not being able to homeschool is wrong, and yes, banning it is a violation of some kind, maybe even a violation of a right, but it's not a violation of a fundamental right, which is a special legal category and not the same as other rights. These people don't deserve the same legal status as real asylum seekers. If they want to leave Germany and come here, let them immigrate legally and go through the same process as regular immigrants, rather than taking spots away from people who will die if they don't get asylum.