I'm well aware of the issues faced by homeless people, and none of that negates anything I've said, if anything, it actually strengthens my point - highlighting just how much more vulnerable and more at risk of being victimised they are. Mentally ill people are at a much higher risk of being the victims of violence than the perpetrators of it.
We're also not talking about ideals here because in an ideal world people wouldn't be homeless, but they are, and they deserve to be able to defend themselves with a weapon of their choice.
Maybe instead of worrying about the "unstable" homeless people, worry about the circumstances that put them there and the people who actively act to harm them (the government, the police, NIMBY's, and so on)? I guarantee that's a much better track to change.