First, "solving" homelessness is wishful thinking - it can't be solved given the modern political climate, because on some level it would be accepting that some people should not have agency for running their own life, and need to essentially be made to comply to a set of rules to prevent them from self destructing. Good luck with that.

The best thing one can realistically hope for is inverting the opportunity cost committing crime the other way, where its generally more favorable not to steal. If homeless people can be shot by security guards for shoplifting, its almost guaranteed that you would see crime statistics reduced overnight. Homeless people would be more incentivized to pursue lives closer to normal.