I just read the sanction thread. I don't actually see any wrong admin behaviour. He didn't say "My kid is OK". He said "my kid understands the rules about language" and "When my kid does something wrong, he gets the same punishment as anyone else".
As you yourself said, your friend needs to learn why bad language is inappropriate here first, so that seems to justify the Admin in saying the first. Everyone there also stated the reluctance of reducing the punishment - that corresponds to the second bit.
He also said that his kid knows he should walk off when angered ... not that he is always able to, just that he knows it and is expected to do so - and is held accountable like everyone else when he fails.
He actually mentions a case where in RL the kid failed to manage that and is dealing with consequences ... where again did you see that the kid "does things normally" ?
It's too bad about the evidence, he should have sent you that if your friend has trouble with it himself. You can still ask your friend to forward the PM to you, or ask the reporting person to send the images to you.
I really hope you will help your friend to manage a fulfilling experience on the server. He will have his second chance when he serves the ban - like everyone else. It is definitely hard when everyone needs to learn the hard way, but that's how it goes around here. (Edit: Yes, it is cold. But it cannot be otherwise in the environment of Internet anonymity, or after this we would suddenly have a server full of autists. At least in sanctions.) I am sure that nobody will hold grudges against him or pass judgement for his condition - and if they do, they will be dealt with by the admins.
Yours is an honourable effort and I sincerely wish you good luck.
TL;DR
* The admin didn't say his relative does OK, just that he has to deal with consequences like anyone else.
* Too bad about the evidence, try to get it elsewhere
* Good luck