Firstly this may be incoherent - long day, DS ill so little sleep, and heavily pregnant! Apologies in advance.
Retro, simply saying "state educated" doesn't seem reasonable to me, for reasons stated. But affording wider access opportunities to poorer kids, who are still attaining well enough (overall, there are 500 places in the county at grammars, and you can select several choices, so the just under 2000 who apply have a 1 in 4 shot of a place at a grammar. You don't have to be a genius) would be a big improvement. They do actually offer a threshold preference ranking system for pupil premium applicants to the top performing grammar now, I've found out today after checking the website for updated info, but the threshold isn't that much below the normal entry, which IMO for kids that low-income could be extended without serious academic impact. And I think that the preference ranking system could be expanded, say to those with parents in receipt of Working Tax Credits, or with a household income below £20,000 instead of the current £16,000. The Rowntree report says that even small increases above the poverty line have measurable impact on performance, so it would be interesting to see if that extension increased the uptake of places by poorer children? It would also be good to see schools have a pupil premium bonus perhaps if they got entitled kids to perform really well in the 11+. Though I think extending the early years interventions throughout primary would be the best way forward. The thing I find most frustrating about the annual politicians' manufactured outrage over Oxbridge admissions is that they're attacking people dealing with the education system's own failings. By the time a kid is 18, you're largely dealing with the educational level they present with. At primary level you can still make worthwhile interventions to raise standards.
The test is very hard to perform well in unless very familiar with the format - so you have an arms race, in which parents who want their kids to have any sort of a shot find the money to get them tutored to the point they can answer in their sleep. (When I say privately tutored, I don't necessarily mean privately educated, I mean private tuition aimed at teaching to that specific test. A prep school that didn't prepare for it could be as wonderful as you like in genuinely academic terms; the kids would still do relatively poorly at the 11+.) That's one of the reasons I think altering the test needs to be imperative. All test outcomes can be improved by good tuition, but to varying degrees. The ones they use at the moment genuinely shocked me, because getting stellar marks so obviously relies upon highly teachable techniques. That should change if social mobility/equity is at all a concern. It's basically setting up a system where you need to know what extras to provide and then have the means to provide them to have a really good chance of success, which for access to state-funded educational opportunity isn't really acceptable.
I think the assumption that making a simple state/private division will prevent privilege weighting the scales is also very misplaced. I honestly don't see why my son should be ahead of anyone in the race for grammar schools, even though he's in the maintained sector. He's not remotely deprived; in actual fact he's getting every bit as good an education as most prep schools could dish out, between home and his Outstanding state primary. If reducing social inequality is the aim, then helping children from the very poorest families is surely the way forward - not giving upper-middle class kids in the state system a head start over their private school peers. I do think if your aim is genuinely to widen access, then state/private is far too broad a brush. 93% of the population are state educated. It is just unreasonable to treat that number as an amorphous group - if you want to boost the chances of bright kids from underprivileged backgrounds, which I applaud, then you need to at least try to identify who they might be, and "state educated" alone isn't going to do that. Privilege is not just about whether education is in the independent or the maintained sector, and I don't see why privileged kids in excellent state primaries should not themselves have to step aside, so a few more kids from really deprived backgrounds had at least a shot at it. I'm not that fussed about kids with very involved, engaged and educated parents - kids with lots of cultural capital. They'll thrive wherever they go. I'm a lot more bothered about kids who start the academic race with their feet tied together. Given poverty alone has been clearly shown to do that, then it seems a pretty good place to start widening access.
Damn I honestly don't understand the point you're trying to make? 
(I've had almost no sleep - DS isn't well). My point is that kids who apply to grammar from prep schools either have parents who can afford to continue educating them privately, or who have amply proven that they're willing to make huge sacrifices for their children's education. I wasn't arguing that state schools with comprehensive intakes are bad, in fact I said in an earlier post that we've decided not to apply to the local grammar at all, despite being pretty sure we could coach DS in, because I'd rather he went to the very, very good state comp near us. I'd rather he went through primary without that level of anxiety on his shoulders - can't see how 2 years of tuition for an exam 90% of kids fail at the tender age of 11 can be ideal for any of them, and nor is the atmosphere in a school that pressured for entry likely to be healthy for many, perhaps even most, kids. I do though very much agree with your point that 11 is way too old to start worrying about equality of educational opportunity. I think interventions need to be made a great deal earlier and a great deal more concertedly. It was one of the things that always bugged me about political outrage over Oxbridge admissions - there is an extent to which Oxbridge pick up the pieces of educational policy, and they can't socially engineer that much when faced with half the qualified kids coming from 7% of the schools. The politicans love that target, because it's the very first point in the system where access/equality of opportunity is out of their hands! It's just too late. I do think extending preschool provision to 2 year olds, if good enough quality, would be a great idea, as well as targeting extra support at infants level. (No idea to what extent that's already happening - perhaps a teacher on MN could talk about that?)
I'd say that many children who recieve it aren't in poverty. They really aren't.
Household income has to be below £16,000 a year to qualify for FSM, or you need to be in receipt of means-tested benefits such as Income Support. The national average wage is £26,000 a year - they live below the poverty line. If you think someone with a child on 16k isn't poor, and that jobless people on benefits aren't poor, then I really don't know what to say to you. 
You're flying in the face of all the statistics and all the data with that statement - both in terms of relative poverty, and outcomes for the children themselves. People getting FSM for their kids are poor, unless they're fraudsters.