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I think that this is shortsighted. Imagine a scenario in which a person might use a Nazi salute in public: it is unlikely that it would be used by a single person just out doing their normal routine. Consider the recent context of Elon Musk, who was at a very large political gathering and was expecting people to respond in kind with their own salutes.
Do you want more visible Nazis? They show up when you summon them. They don’t go away when you outlaw their symbols, but they do find it less appealing to promote themselves online or have rallies to recruit more would-be Nazis. I understand your hesitance to let the state do these things, it’s a large part of the reason why I don’t believe in capital punishment. But if the state doesn’t have the power to stop this peacefully, the next step (no matter what “side” is doing it) is violence.
The comparison to saying slurs is just not equivalent whatsoever.
They probably mean stimulant medications for ADHD, like Adderall.