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).
You need cars because you live in the country where cities are built around the assumption of car ownership. Go to any place with an adequate public transit and you'll see a lot less cars. Go to Tokyo and you'll be shocked, it's the largest city in the world and their highways are practically empty. It just doesn't make any sense to drive when your public transit is that good.
I have been to Tokyo, and I agree with you; perhaps I should have been clearer, but my point was that in American cities you DO need cars (this is a given, however unfortunate it is), and so you are going to already have a car and the costs that go along with it, and as such you are not going to use public transit even when you could because you already have a car. If I had had a car when I was in Tokyo I certainly would have used it! Public transit takes half an hour to go a couple miles, and cost wise gas is cheaper! In the scenario where you already have a car, it doesn't make sense to use public transit 90% of the time. Does that make sense?
The only point I was trying to make here was that cost isn't the reason public transit doesn't work in American cities, and making public transit free will in fact make it worse rather than better. I want good public transit too, don't worry!