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Do those families, with legitimate grief, anger and reasons to be vengeful, turn to extremism?
That's the tendency. It's a tendency promoted and exploited by the radical right to advance right-wing extremism generally. Far from deploring incidents of violence, the radical right in fact finds them very useful. This is why the radical right is often found to have fomented such incidents themselves in 'false flag' operations.
Typically the radical right conscientiously avoids any thoughtful, just, and effective approach to ending the violence in favor of an approach which instead escalates the violence so that it can be exploited by ambitious demagogues and weapons manufacturers.
The techniques for accomplishing that exploitation have been well-understood since at least the 19th century, including examples of the misinformation and disinformation techniques so often seen posted here.