I think the truth of the matter is that, despite your son having a father, and despite whatever legal custody agreement the Court has imposed, your ex husband's alcoholism has progressed to the point where he can no longer function as a responsible parent. It isn't just that your son "feels" responsible; it is that your son IS responsible because his father is now so impaired that he can't be a parent.
Because your son is 17, and can cope with some of his dad's dysfunction, your ex husband's incapacity has been shielded from view. If your children were younger - toddlers or elementary school - his inability to properly and safely parent would be glaring, and I would guess you would have been back in Court to take away his rights as a parent.
So, it is a strange outcome, but you are now essentially a single parent without a partner. That is a different form of parenting, and a difficult one, but necessary. I raised two children alone, and at times them and three step-children while my husband travelled, and worked a stressful job an hour's commute away, so I understand and sympathize. It is complicated to provide child care, and in your son's case that seems to primarily mean transportation, but it is necessary.
So, Liz, I'd suggest rethinking this situation as if you had sole custody and no help whatsoever from his father. That's the truth of this situation, and I still think that letting your son bear the emotional brunt of his father's failures is too much for you son, and you should protect him from that.
There are lots of ways to solve this without requiring your son to stay at his father's home. Maybe this is an expense with Uber you will just have to swallow for a while. Can your son get his license any sooner? And as for the car, your XAH is using that as a bargaining card with your son, and that's a familiar and painful reminder of how he used to treat you. I'd say, just plan on getting your son a car yourself, and end his father's control.
It is hard to be a single parent, but there are many options. Sometimes people from the church will pitch in and drive. Or there may be an older teenager or college student you could pay to drive your son. Your son needs to be free of having to parent your XAH.
ShootingStar1