"Birmingham appeal successes seems to be extremely rare in comparison to the successes in other areas eg Bucks."
In areas with a state-operated 11+, presumably appeals are held by the education authority, just as any other admission problem. Whereas in Birmingham, the appeal is to the foundation who operates the schools. The authority has no skin in the game: they're operating comprehensive schools, not a binary system, and although they still voluntary aid the KE schools, the admission policy isn't really much to do with the LEA beyond compliance with the national rules.
The foundation make it quite clear that admission is on test results: not the test results as a proxy for some other ability, but the test result itself. If your child was ill on the day, arrangements can be made to take it at a later date. Otherwise, unless you can show that they failed to follow their own rules (and you almost certainly can't), it's hard to see why they have any incentive to do other than point at the test results and say "well, there you are".
Although KEHS takes a report, I don't think the voluntary aided schools do. As fm explains, the pass rate out of the overall population of Birmingham is low, and the chances of a particular school being able to make "would pass // wouldn't pass" judgements with precision are equally low. Why would a primary school head be able to, or indeed be expected to, make that judgement? How would it be standardised? As fm again points out, G&T is defined to be the top 10% within a given school --- there's even a G&T programme at KECHGS, when in any other school every child would fall into that category --- so being on a G&T programme in your primary school means little at a city-wide level.
And anyway, the foundation has made their position clear: it's the test, not the test as a proxy. Unless you can show a clear reason why you couldn't take the test or the re-take, that's pretty much the end of the conversation.
My hands are dirty, because I have two children at KE schools. But Birmingham has the worst system of all: a voluntary, high-threshold 11+ for which only children whose parents make plans well in advance have much chance of passing. The children who get into the KE schools would all, without exception, do well at comprehensive schools because they have impeccable parental support, strong primary education and a lack of special needs (because the KE schools pretty much ignore statements when considering test results).
And in a vicious circle, it's the absence of those children and more particularly their supportive, engaged parents that hampers the comprehensives. In the 1970s, there were a lot of academics whose children went to the better comprehensives (in those days, for example, Shenley Court was a shining city on the hill) and their results were within a grade or two of their contemporaries who went elsewhere (my two "as bright" friends at primary went to KEFW and KES, while I went to Shenley: we all got the same grades at A Level plus or minus one, and went to directly equivalent universities). In 1976, everyone was so glad to see the back of the main 11+ (I was the second cohort not to take it) that the comps were able to be genuinely comp --- everyone I knew who went to grammars were younger siblings whose older siblings were already there, whereas I am the eldest in my family.
But over the years, that solidarity (for want of a better word) has leached away, and the middle classes are back into the grammar schools and hence out of the comps. Maybe the children do better, although I'd take some convincing. The music's better, for sure, because there's a critical mass of the middle classes in one place (although the demographic changes at KECH mean that won't necessarily remain true in ten years' time). But the middle classes have re-created for themselves a past world, where a high-stakes exam with uncertain outcomes is all that keeps you from schools that in some areas are secondary moderns in all but name. But now that exam is higher-stakes than it was before, and much harder to pass.
For any given parent, the best strategy is to get your child into a KE school: you have to be a lot more politically committed than I am to opt out as people did in the 1970s. But the effect on the city at large is negative, because motivated bright children and engaged parents now have no stake in their local schools. It's a classic example of the Tragedy of the Commons: the rational, moral position for an individual is not sustainable, but it would neither be rational nor moral for them to do otherwise.