He a 16 year old who has lived with an abusive father. He's a child and shouldn't be having to pay board, that is unfair IMO.
You are now seeing him as his dad which is also really unfair. I wouldn't throw 'gas lighting' around, maybe he thought he had told you, it seems you're judging him by his fathers standards which is really bad IMO. He is not his father but he has been through a lot.
He very clearly needs time and support that he's not getting at the moment. That's what he means when he says you are always too busy. Is he at school/college? He hasn't got his dad so he really needs to know you're there for him to help and support him, he's just a kid.
You need to remember that he is a 16 year old child that has been through a lot and is now struggling. Obviously you can't get him driving lessons and a car if you can't afford it so you need to be suggesting he save up from his part time work for that - not to pay you board.
He's lost OP, it's not clear if he's in school or college but if not then he needs even more help and support. It sounds like he didn't have the confidence/self esteem to apply for apprenticeships without some help and support - which he didn't get.
Mine has an apprenticeship and got a lot of support to get through the different stages, writing a cv/cover letter, how to answer the personality tests, what he might be asked at interview and how to answer based on his experience. You need to start supporting him if you want him to get anywhere IMO, he's just a kid.