Tich
I was the hellish teenager you are describing, my father threw me out on several occasions. But please listen to me. I grew it out of it. One day the fog lifted, the hormones subsided and I became 'normal' again. I can not believe now I did those things.
My mother did stand by me, throughout it all, and I have never stopped being grateful to her even now thirty years or so later. Even if you can't have him at home at the moment, he can still be told he means the world to you, and that you are having a break because you are ill and worn out, but at some point you hope he will come back.
Will you consider taking him out for dinner or a film in a few weeks when he has settled down? Tell him you love him, the door is always open and that you are, as you have always been, will be there for him.
See it as temporary change, it does not need to be permanent. Perhaps you could say to him that you would welcome him home at some point, so he does not feel rejected and still feels he has choices.
He is cold and distant because it is hard to deal with, even he doesn't show you, he DOES love you. Leaving home is a big deal for anyone. He is going to his father's and not into care, or on the streets, he will be safe and hopefully you will have rest and time to get better. I think you will both be in a better place soon.
After sixteen years of love and care, you could never say you have failed him, to me you sound in desperate need of a rest and a break, you can have that and still love him dearly, and hopefully in time you can repair your relationship, as I did with my parents.