If you can reliably kill an unarmed woman in the street, there is no need to do so. State your threat, which is now a promise, which will achieve compliance with your aims, without actually having to kill anyone. (On average.) Indeed carrying out the threat will be counter-productive. (On average.)
Killing people unreliably is chaos. Chaos is - get this - not order.
Do not feel sorry for this woman. She was not your friend, she was certainly not your vassal, and she was not your responsibility. Her life or death is meaningless to you and should be treated accordingly.
That the perpetrator thinks he was a neo-nazi does not make him a rightist, even if you accept nazis as right wing, which I don't. Crazy people believe crazy things. At best you'll find he was to neo-nazism as a lapsed Catholic is to Christianity. More likely it was a loud statement with little to no grounding in behaviour, taken to justify a pre-existing desire to murder someone.
I'm skeptical of the claim it's a propaganda win for anyone. Everyone who was friends with the woman, and thus likely to think badly of her killer's group was already not even relatively right-wing. As for the other side...look, humans are murderous. The idea of killing the inconvenient comes naturally to us. Nobody sympathetic to the other side is going to be beat up about her death, and thus unlikely to grow in antipathy to other side.
What I'm seeing is a massive waste of time all the way around the table.