If you're concerned about the burden on social services, you should demand that your local government provide no-questions-asked, no-strings-attached SRO apartments for anyone on the street. It is far more cost effective than constant EMS and police attention. And despite your claim that "vagrants" will not accept organized help, I guarantee that almost all of them will if you remove the arbitrary and unnecessary rules and restrictions. There's a reason that many european nations have effectively no homelessness, and it's precisely because free housing is a given.
If our society treats getting someone into an apartment as a <i>goal</i> that we need to work for rather than a <i>favor</i> they need to work for, it becomes much easier.