>Murderers aren’t really representative of the homeless.

Nobody is arguing they are.

>Bear in mind mere homelessness is not unequivocally a crime in the states

Nobody is for prosecuting homelessness. Vagrancy, loitering, public consumption of drugs and alcohol, obstruction of public right of way, public nuisance, public defecation, trespassing etc, etc, are crimes though and we should be going after their perpetrators regardless of their having a home.