Firstly, I would say that most secondaries nowadays have very good pastoral care. If she is good at English and Sport she is already likely to find a niche at a state secondary.
Secondaries often have loads of clubs that suit children who find it difficult to fit in(drama, chess, ICT, art, specialist sport), they also have the benefit of having dealt with many many types of children and needing to support them through various problems. They have the expertise that primaries often lack. Or is that just my imagination???
Ds2 has a diagnosis of ASD, and bad at sport, bad at making friends and academically quite behind, yet is already enjoying his comprehensive in West London (although it does have an Outstanding status). The SEN dept have identified a lot of his difficulties and is intervening, without any statement. He is on SA plus. He finds school very enjoyable, has the odd day when it is all too much (two refusals since Sept), and hates the homework, but we are surprised how much he does enjoy it and copes independently. We did not even have this school as our first choice as it wasn't deemed to have the right SEN expertise, yet it has proved very helpful.
Of course as you say, there is always the doubt that another school might be a better fit or more nurturing, but I would talk to a few parents whose children attend these state schools and find out how their children fare, what they like and dislike about the schools etc.
With regard to your daughter having unrecognised problems, I would go to your GP and ask that your daughter be referred to CAMHS for an assessment. You are allowed to self refer, although if the school were to list her "problems" you are more likely to get an appointment. Our ds2 was thrown out for referral, and it took us two years to get a diagnosis of ASD, which is now pretty obvious to all who meet him.
Our ds1 who has only been recently diagnosed with dyspraxia, bad at maths, bad at sport, difficulty with friendship, distracted, was never flagged up at school because his classroom behaviour was fine and he was a "lovely" boy. He should have been referred long before that, so it is completely possible that you are NOT OVERREACTING. So quite aside from the schools thing, just get the ball rolling so you can rule out any underlying issues. When we finally took him to an OT aged 13 we found he was in bottom 2 percent for motor skills, yet had successfully gone through primary without any concerns, or any IEP. I think the first term of secondary would have been a lot easier for him he had been known to have dyspraxia/organisational problems and the SEN dept had been proactive. His behaviour did deteriorate at secondary because there was no formal recognition of his skills "deficit". He loved school, and still loves it though!
sorry long post, just TMI!