Whole lotta assumptions here. The biggest one is "Greater police presence is an excellent deterrent for homeless misbehavior and crime."
So much of what you said is, honestly, what a lot of homeless people need to do, just to survive. When you don't have safe, ready access to bathrooms, you go outside. When you don't have a job and can't get money, you steal to eat. When you are subject to this misery for long enough, you stop caring about what polite society cares about.
Police presence sure does deter this behaviour, but only where police are present. It doesn't _stop_ this behaviour, because it can't. Cops can't fix homelessness, they can only move it along somewhere else, turn it into some other neighborhood's problem.
If you really want to fix the problem, look at you and me. We don't crap outside, we don't steal, we're not making neighborhoods scary and dangerous (I assume). And why? Because our basic needs are met. We have food, water, shelter, sanitation, entertainment, employment. If you really want to fix the problem, THAT'S the problem you fix. Sadly, social programs for the homeless ("handouts") aren't nearly so politically popular as spending that money on policing, the legal system, and the prison system, despite being many orders more efficient.