I was that kid too but slightly different circumstances. At 14, my mum was very strict and I was rebelling, she didn't have much cash and used to kick me out often as punishment. One day she wouldn't let me back in and I ended up in a childrens home - something she would threaten me about. I ended up liking it there as I had freedom and finally money for designer trainers and although I didn't realise it at the time, my mum was depressed and a which meant I didn't enjoy my home. Social services attempted to send me back as my mum wanted me back and they obviously wanted to save money - I rebelled and ended up back in care. Also realised that I'd get a flat at 16 if I went into care again.
I got the flat, and over time built up a better relationship with my mum. Obviously she was hurt and I feel bad for putting her through that but it wasn't a healthy environment and I was very rebellious- got arrested and used to smoke cannabis also.
I'd let him go. If he comes back, it won't be because he wants to be there and likely his behaviour will be worse. I'd sit him down and say you'll give him what he wants but these are the conditions, he sticks at education, he comes around for Sunday dinner and an extra evening a week and whatever else you want to chuck in that you feel you can negotiate.
He has to get this phase out of his system but it has to come naturally and not by force. I fixed up myself within a year and by the time I got my place at 16, I had a job and was sensible because ultimately, I grew up in a home where we were taught how to behave, I was just going through stuff and my mum wasn't best equipped to handle it.
Your relationship may improve by being apart and it means you can focus on your younger kids without the interruption of police and the other drama it brings. It doesn't mean you're giving up on him .
The only thing about care and group homes is that they are able to run wild. I did for a bit, it was fun and crazy but I then realised that I didn't want my life to end up like the other care home kids. Who knows if he will have that realisation. Of all the kids I was in care with, a couple have been murdered RIP, a few have numerous kids and no career/job/focus in life, one is a prostitute and a few are in jail/have been, some are druggies - not many make it out a success which I'm sure won't make you feel better about he situation. I think having the regular visits with him may slowly show him there is more to life than freedom, running wild and free stuff although at the time, that's all they consider important.
I wish you luck. It's not a reflection on you as a mum. My mum tried her best but it wasn't enough to keep me tame. Again, you're not giving up on him, you're just not fighting him, you're working with him and letting him know he can always come home. I'm in my mid 20s now and have a much better relationship with my mum which I'm not sure would have happened if I continued living there.