Focusing on your last sentence.
Often, homeless are rounded up eventually if they make a nuisance of themselves (aggressive panhandling for example) and they spend the night or two in jail...paid for by your taxes.
If they are found to be mentally (and many are), they go into treatment...paid for by your tax dollars.
Now, I am willing to bet that should they show up (or perhaps already have), you want them out as they bring down the neighborhood value and more importantly (I'm being serious here) they could be a safety risk...and you will call the police to remove them (I would as well, and I have)...so regardless of where they wind up....you and I will foot the bill with whatever happens.
So...since we are going to pay for it anyway...why not pay for programs that gives them housing, a treatment plan and a stable life a roof could give them to help them get jobs and to be more productive members of society instead of paying for the vicious cycle to continue forever. It may take time and not all will be successful...but since we are going to pay anyway, shouldn't we at least try?
That's for those who aren't mentally ill and who...choose...this lifestyle. I'm not sure why choosing this way of life is appealing to a rational person to live on the streets, in filth, dealing with violence, and I'm not sure why you think that way. But the sad truth...for those of us who deal with homelessness more than others....it's usually related to drugs and mental issues...not really a desire to live off the government teat by giving up the amenities of an apartment, electricity and available food and hygiene and living like a grub on the street.