Having a safety net and more homeless services certainly does not make anyone "want" to be homeless, but it does attract homeless people from areas of less support. I personally know of people who have literally been given a greyhound ticket to SF from e.g. Michigan[1]. At the same time, the roots of much homelessness, e.g. issues surrounding mental health, drug dependency, etc. are not solved with housing availability and clean needle deposits.
Not sure what the solution is.
[1] I'd like to see the numbers people have researched, as anecdotes do not make for good data.