Oh Rosh I have heard those words so many times from other carers "I feel so guilty and responsible for this child" and talking of being a "failure". Your feelings are real, but you have no need to feel guilty, and you are not responsible for this child and most certainly have not failed. I think the most telling phrase in your post is that the child is destroying us - that says it all really. You have to call time, because his behaviour will get worse rather than better.
The failure lies with the LA who have PR for this child not you. Their excuses are inexcusable. You must be very angry and frustrated.
I would also say that the placement with you is not right for the child either. Please don't take this the wrong way, but I suspect you may well be "too nice" and have gentle dispositions. A child like x is going to take advantage of this (not consciously of course) and his out of control behaviour will escalate as it is doing at present.
I actually started the teenage placement scheme I mentioned in the area in which I worked when I was a social worker. I took a risk and recruited some carers that others perceived as "a bit rough and ready" and lots of them had been in trouble with the police when they were younger and even been to prison. I had to ensure that these incidents were over 10 years in the past. I didn't set out with an idea of the type of carers I wanted, but I knew what I didn't want, and that was nice, ordinary people who would make good foster carers for younger children, because I knew they would not cope with some of the kids we had in children's homes at the time, and that these kids would, to use your word destroy them.
SO yes some of the carers I recruited had tattoes and swore and lived on the local estates, and stood for "no messing" but more than anything they seemed to have an understanding of these teenage kids, and I would be half way through my speel on my initial visit and then realise that it wasn't necessary, because they seemed to "know" these kids. One carer had been in care herself and was in a children's home, which she hated and was constantly "running away" and she talked of wishing there had been a teenage scheme when she was that out of control teenager, and now wanted to give another teenager the chance she never had. Another couple had lived on an estate in another town which had an appalling reputation (a real sink estate) and again "knew" the kids we wanted to move out of children's homes.
The scheme was an absolute success, not to do with me, but because of the carers, but I don't mind taking credit for knowing the type of people who would cope with these kids. I did a prep course (like fostering) but obviously focussing on teenagers and there were a few people who dropped out which I encouraged because I wanted people to "self select" though I did have the "power of veto" if someone was totally unsuitable. The first cohort were of 8 carers (all couples as it happened) and those people were absolutely brilliant. My intuition had been right. I remember one lad was terrifying the female staff in the children's home and had one member of staff up against the sink and was threatening her with a broken milk bottle. The other member of staff apparently was too scared to do anything about it. I wanted him out of the home and placed on the scheme. I was totally honest with the couple (the bloke had been in a fair bit of trouble in his younger day and had skipped most of school and his wife was one of 7 kids who had lived on the estate all her life and they had 2 kids of their own though not teenagers. When I told them about the milk bottle incident I thought it would put them off but NO - the female carer said "Well if he comes here, he won't have me up against the sink" and I looked at her and thought "No you're right he won't" - (she was about 5' 3" and slim build) but it's not about size, as kids instinctively know who they can mess with and who they can't, so I placed him with the couple and there was no trouble whatsoever.
There was a sw colleague who again was small and slim but she was a black belt in Judo and she explained to us that keeping yourself safe from attack was nothing to do with size, it was about your thinking or as she put it your head - she did some classes for us but I knew that my head would never be in a place where I would be able to cope with violence.
I'm taking a risk telling you all this I know, but I am hoping you will understand what I mean. I am definitely NOT criticising you in any way shape or form, and really do think you need to get your notice in. That will bring the sws round, offering this that or the next thing, but it's too late and you have to be resolute. They may try the "well we have nowhere else he can go" (tell them firmly if they use this despicable excuse, that this is their problem, not yours) In your OP you said you were "torn, stressed, miserable and knackered" and then that this child is "destroying" you. You will need to think about what you tell the boy (I am assuming it is a boy, not sure why) and whether you want the sw there when you tell him. He probably won't show any emotion and will just shrug it off, as he will never have learned how to deal with his emotions, sad but true.
Sorry I should probably have PMed this post. Incidentally do feel free to PM me. Quite a few foster carers do this.