He does sound like he is very unhappy and you have done everything 'right'. The stress of constantly worrying whether a child is going to get bitten must be incredible - and even if he was rehomed, would he be any happier, and would you ever be able to stop wondering if he had bitten someone?
Years ago, we had a dog like this, too. We have brought a number of dogs up from being pups and I've had dogs my entire life, and had never had a problem. We kept one of my own dog's pups - she was the best natured, sweetest, loveliest dog I have ever owned. Absolutely bombproof. And the person who bred her said she bred for temperament (as all breeders do - I have grown cynical about that claim after what happened).
Pup was unusually aggressive from the start. He was stunning and beautiful and his mum was Crufts qualified - he could be a top show dog, but I was allowed to have the pick of the litter, so chose him kept him as a pet. As he got to a few months old, it seemed clear something was wrong. No-one ever shouted at him or mistreated him in any way but like your JRT, there were things that scared him. I felt like I'd 'ruined' him somehow and yet, realistically, I have always had lovely dogs and he was brought up identically to them. He was especially terrified of going in cars - despite us doing, as we always do with our dogs, everything right - taking him in the car from 8 weeks old at every opportunity, etc. One day taking him to the vet's, he bit my son and the bite was really vicious - not a nip but it would have been a sustained attack if I hadn't been able to pull over and deal with him, literally prizing him off my son. We still kept him for a while after that but he was only happy in his crate, attacked my other dogs and made their lives hell and the kids obviously wouldn't go near him. He hated everything and the aggression seemed like fear aggression.
Finally, had enough and rang the breeder to see what she'd say, hoping she'd have him back, if I'm honest (in retrospect I regret not having him PTS). And when I told her about his constant terror of everything, and aggression she literally just laughed and said "Oh he's just like So-and-so" (big show winner so bred from extensively) "That was his grandad - he was just like that!"
So much for "breeding for temperament". I realised it was a fluke my own dog was such a sweet natured thing. She had the genes for something else entirely. Anyway breeder said she'd take him back and implied it was probably my fault - he needed more training etc. She told me later he tried to kill her neighbour's cat, one day. I had a cat all the time we had this dog and she avoided him but that could have been her...
Next thing I knew, she had an American breeder/trainer staying with her who had fallen in love with this dog and she told me the dog would do anything for him. So he now had him. (I suspect she sold the dog, too). This dog went to America, won loads of shows and that was the last I heard. Then a couple of years ago, someone told me that the dog's career was spectacular but short as... he bit a judge in the show ring. He was PTS.
Years have gone by and I now know I should have had that dog PTS there and then - because he was probably bred from, even though they knew his temperament issues. And they would no doubt have had him PTS themselves if he hadn't been so stunning to look at.
I know those circs were unusual as this was a show-dog. But the personality you're describing is exactly what we were contending with - a fearful, aggressive, unhappy dog. My own dog, his mum, was the best dog I have ever owned - and he was a nightmare. They say meet the mother and preferably father too when you buy a dog but that doesn't really tell you much if, like happened to you and us, those dogs are perfectly normal and fine you can't guarantee what grandad was like, or great grandma, or whose genes are dominant...
In other words, some dogs are born that way. You can't blame yourself in fact you have tried everything.
I felt very guilty but when that dog went back my other dogs were instantly happy again and the positive dynamic restored - and if he'd been my kids' only dog it might have put them off dogs for life but luckily, like us, you have another nice dog.
I regret not having that dog PTS. I took the coward's way out letting the breeder deal with it but in retrospect, she wasn't qualified to deal with it as she had made the problem in the first place. You must be kind to yourself and let this dog go now, if the vet has already eben consulted and there is nothing physical. From what you describe - sounds awfully familiar. I just concluded there was nothing we could ever have done and within a year or so, he was PTS anyway.