One of the reasons why the first German democracy (1918-1933) failed (and gave rise to Hitler) is because the German society was very much compartmentalized and split into "subcultures" at the time. In fact, large parts of the military chose to continue as "para-military" units after the end of WWI. These violent groups that saw themselves as above the law played a significant part in Hitler's ascent to power.
When the current Germany was designed after WWII, many safeguards were built into the system to ensure that there would be no more "parallel" societies. Two obvious examples are 1) the universal draft (flow all parts of society through the military to prevent the military from isolating itself) and 2) no home schooling (the public schools serve to promote democracy and western values; the public school system is designed to be the "melting pot" that integrates immigrants and fringe groups).
How would you feel about this case if the home schooling did not involve Christians (Note: I'm not religious myself.), but radical Moslems? What if it were about Nazis indoctrinating their kids with hate? What if it were about a sect that raises their kids to prepare for a collective suicide?
The German (mainstream) belief is that the government has to step in in these cases. It has to protect the kids from their own parents.
The family that was granted asylum may very well be fine parents and teachers (I don't know them), but for obvious reasons you can't have an effective law that says "you may only homeschool kids for Christian reasons, but other choices are not ok".
So, the German society, under consideration of its history, chooses to err on the safe side: no homeschooling. Period.
Of course, the particular circumstances may very much differ for other societies, so that other societies may reasonably come up with different rules (e.g., the US). But that does not invalidate the reasoning behind the German laws.