I began to reply to this last night, but then found I was getting a bit bogged down, because I didn’t really understand everything about your post. So I think I’m going to have to answer with some questions, as well as comments, if that’s alright?
Firstly, I didn’t understand why the title of your post was regarding ESA non-contributions. I assume, by that, you mean non-contributory, or income-based ESA? But if you’ve been working, and your NI contributions are all up-to-date, then surely the contributory version would be the one that applies to you? Or are you looking ahead to the one year after which that would run out?
Although I don’t think you did wrong to appeal (it’s always worth a try), I don’t think you have much chance of success on the grounds you have chosen. You would need to appeal on the grounds that the 0 out of 15 points was wrong, and you are NOT fit for work. The fact you are still employed, and therefore not in a position (you say) to look for work has nothing at all to do with your fitness for work, so I don’t see how you can dispute the fit-for-work decision, by saying you are not available to look - which is a totally different thing.
I must admit, I’ve not really understood why you couldn’t look for other work, in any case. Do you have some specific clause in your contract that says you’re not allowed to, whilst still technically on the payroll? 20th August is less than a month away, after which they will have no claim on you. In the current climate, if you began looking today, do you really think you’d have landed a new job and be asked to start before the 20th August, meaning it would conflict with your current contract?
Even if it did, I can’t see that your present employers could do very much about it, other than stop paying you for the overlap, because you’d already found something else. Or will you collect an ill-health pension, that’s conditional on not being able to work at all?
If you have been found fit for work, scoring 0 out of 15, then unless you can get that overturned, I can’t see how you can get ESA at all, let alone “top-ups” and extra help to find something else. They will take the view you don’t need top-ups or extra help, because you don’t have a problem that limits your ability to work. I’m not saying that’s true, but that’s the harsh reality about how they will treat it - that you are an ordinary person, with NO health issues. Therefore you might well be able to claim Jobseeker’s, but not ESA (unless, as I say, you can get the fitness ruling overturned). It’s probably too late now, if you’ve already submitted your appeal, but you really do need to focus on the assessment itself being wrong, and not on non-health related reasons why you’re not available for work.
I think (not absolutely positive) that if you apply for Jobseeker’s you may be able to access a “disability” advisor, who would be able to assist people like yourself who don’t qualify for ESA, but nevertheless don’t have completely zero obstacles.
I looked into this (briefly) for myself - but not that recently, which is why I’m a bit vague. I was interested in speaking to an advisor, but discovered I’d have to be claiming benefits first. Which was a bit of a catch-22, as it meant I’d have to prove I was actively looking for work. I wasn’t ready for that, as ideally I’d have liked to chat to the advisor first, about my options, before saying yes, I’m available and looking. But you can’t do it that way round. So as a result, I’m not claiming benefits, but have no access to any advisor either. I really can’t see why you should be prevented from speaking to an advisor, simply because you have chosen not to apply for benefits - yet. But that’s a whole other story, and doesn’t have much bearing on your problem. I just thought I’d mention such advisors do exist, but you’ve got to be claiming first.