There are other things to consider. Free public transit would make it more desirable to not use cars as much reducing congestion, accidents, pollution, energy usage. The need to build less high traffic roads may be a money saver to offset. Also it will get more people to exercise by walking to bus stops. You'd need disencentive for abuse but you can personally only add 16 person hours of usage per day so it is not like say free unlimited money or electricity where a single user can add a lot of strain. And spending all that time on transit is a cost. For homeless people yeah you may need to solve that problem! Build more housing for sale and rent driving down prices and reducing the number of homeless yet productive people (with jobs but can't afford rent). Build shelter for homeless with problems they can't hold down a job.

Why do you need free public transit for this? Why not just cheap public transit? Cars certainly aren't free, but people drive them because cars are just far better in most ways; you get to listen to your own music, they can be fun to drive, are incredibly flexible (can go anywhere), don't have the last mile problem, take like half the time unless it's rush hour, etc. Cost just isn't the reason people aren't using public transit. People buy cars because you NEED cars for many other reasons (most of them mentioned), and then once you have a car and pay for insurance and everything it's simply nicer and faster than public transit (rarely cheaper, even only factoring in gas and per mile depreciation, although the BART is so expensive it comes close). (FTR: I'm not sure where I stand on public transit subsidization; I think some is definitely good, but not sure how much I think is optimal).