Even if they did care about the situation of their family members the problem was that many of these children or close relatives were already in institutions before the killings started. The handicaps, disorders, or other conditions were already known to the authorities way beforehand.
The most famous example is Aloisia Veit, niece in the second line (daughter of a cousin so) of Adolf Hitler himself diagnosed with Schizophrenia, she was killed as part of this idea. It is unknown if Hitler knew that she was killed but most probably he did or at least could have suspected it. http://m.zdf.de/ZDF/zdfportal/xml/object/5907354
Another prominent example is Magdalene Maier-Leibnitz. http://www.ev-kirche-esslingen.de/geschichtliches/euthanasie-in-esslingen/
She was born in a prominent family and diagnosed with Schizophrenia but she was also killed, don't know if their family members were hardcore Nazi's, but many Germans agreed in this time that their 'handicapped' children had to be taken care of in specific institutions, where they were eventually killed, so even for prominent Nazi's this was very difficult to stop in.
Lets just say the Nazi's were crazy extremists.
Edit: I did a bit of extra research on this topic before and there is this interesting Book, if you are interested, this article in a prominent German newspaper elaborates on it: http://www.welt.de/kultur/literarischewelt/article114390776/Wie-Eltern-ihre-Kinder-in-den-Tod-schickten.html
The Book deals with the 'sensitive' question if Germans approved of what happened to 'handicapped' people. Among historians there was a discussion if Germans actually knew what would happen to handicapped people, their own family members, prominent Nazi's and other Germans. A long time it was said that they didn't know, that the Nazi's tried to silence it by telling the people their 'handicapped' family members died of illnesses, for example.
Although it is true that the Nazi's did spread these kind of message historians nowadays think that the Germans did know what would happen to their family members [certainly the hardcore Nazi's ofc] and that they approved of it. The historian Götz Haydar Aly mentions in his book the following poll among the parents of 200 handicapped children in Germany around 1920 (even way before the Nazi propaganda regarding handicapped people would start).
The question was if the parents would agree with a painfree ending of the children's life:
The following years the Nazi's would make much propaganda stating that handicapped people should not be allowed to reproduce and even live in Germany, a sentiment already present, but now strengthened.
In other words, the Nazi's approved of these kind of measures, although it was not mentioned in public.
Prominent opposition came from the Church, Christians, who had Religious reasons to defend the Rights of 'handicapped' people but it did not prevent Germans from letting 'handicapped' people, family members or not, being killed in these days.