1. You're clearly NOT incompetent. If you were, you couldn't possibly hold the web and mobile platform together entirely on your own for that long AND handle every on call incident.
2. The thing that surprised me most about burnout was the physical symptoms. You've got them. It only gets worse from here if you keep doing what you're doing.
3. It makes perfect sense to me that you don't feel like you're getting better in that environment. The rockstar developer is a myth, but the band is not. You get better by practicing with other talented people.
4. You already know this, but you're being taken advantage of.
So given all that...
1. Everyone telling you to take a break is correct. Go outside and look at trees, read a book somewhere scenic, shut off your phone. If you need to take sick time to do this, so be it. It's not going to make you feel normal, but it will give your wildly overworked brain some relief.
2. Hit up recruiters, hit up whatever network you have (including old acquaintances). Your story is incredibly straightforward: you held the line on these systems by yourself for years on end, including all their on-call needs, it's enough work for a 4-person team, and no help is coming. You're looking to work on an actual team where people actually learn from each other.
3. If possible, take some time off between the old job and the new job just to take care of yourself.
Good luck.