Look at January 6th. Many people came to "peacefully protest" the election and got swept up in doing something they didn't expect to do.
I don't know if you can conclusively prove they were directly linked, but we've seen how online people get radicalized by finding a community that espouses abhorrent ideas and then go on to commit crimes in service of that community. Real life gatherings and protests can have the same radicalizing effect in certain cases. That's not to say all of them do - the ideology behind the protest matters.
I agree that focusing on death threats would be a better approach - the problem is that it's very hard to find perpetrators and very easy to commit. Deterrence doesn't scale in that case.
I tend to think focusing on the threats themselves would be the better approach. Having trouble seeing how shutting down protest would help.