Much like politicians, diplomats should be held accountable for the actions of their family members, especially their children. In the case of a vicious and unjustified physical assault, the diplomat should absolutely be removed from their position. If they can't properly manage their family, they can't properly represent their country.
Diplomatic immunity is important as it assures diplomats and others on official duty to complete their mission without the interference of the host countries law enforcement. Without it, a host country could charge a diplomat or their entourage with a crime using their freedom as incentive to reach a political gain.
That being said, the key word in the question is "should". As a diplomat, you are there as a representative of your country. If you break the laws in other countries, it makes your country look bad not only bringing dishonor to the diplomat but the country they represent. Because things like this could jeopardize negotiations between the two countries, it is in the home countries best interest to keep a leash on their representatives and strip the person(s) of their diplomatic status as they see fit.
There are two good reasons why a country may not want to strip their diplomats of their privileges. The first being, the possibility that charges against the diplomat and their entourage are false and that the host country may be using the charges as leverage in the agreement between the two countries. The second being that many if not most of the negotiations that the diplomat is there to perform are time critical and sending a replacement diplomat may not be an option.
In the end, it is, and should be, up to the host country or the sponsor to decide if their people are striped of immunity and not up to the host country. Just because they "should" does not mean that they would be required to.
I don't think diplomats should automatically lose their positions if their family members commit a crime, but those criminal acts shouldn't be covered by diplomatic immunity, and it depends on the situation. Minor crimes or so-called victim-less crimes are one thing, but those teenagers committed assault and left that boy in a coma. There's no way they should be protected. If there are no consequences for the boys, then yes, the ambassador should lose his position. I'd assume that with such negative publicity he would be recalled anyway.
It was their family member not them. They likely had no control over it. They can not loose what they worked so hard for because of their family members actions. They might be a good politician and help people so firing them over what their family member does is silly.
The actions of another person should not affect the status of a diplomat. Resolving issues with crimes committed by the family of a diplomat requires a revision of laws regarding diplomatic immunity. If a diplomat and his family members is subject to judicial law and prosecution, such issues would cease to exist. The perpetrator should be punished, not the diplomat.
A professional should be evaluated based on his or her work, talent and abilities. Wrongdoing on the part of a family member in no way negates the work of the diplomat. The son should be responsible to answer for his own actions without his parent's position being a factor. To punish a parent for the crimes of a child is unfair to the parent.