I believe that the most important years in education are from reception to Year 6 and even if it was my intention to send them to a state secondary school I would send them to a prep school to get a solid foundation. That is not to say state schools do not give a solid foundation.
Now before G55 tells us about the awful private schools she's worked in
I do understand that all independent schools are different. I have had experience of both and I have even spent a day at one of the best state primaries in a certain county and I also used to listen to children read in another.
Spending a day in the state primary school made me realise the amount of time that is wasted due to class sizes, discipline and undiagnosed learning difficulties. In an English lesson lasting 40 minutes the children did 10 minutes of work due to the amount of disruptions. The most able table was the most disruptive as they were so bored and the teacher's time was taken up with the least able.
My experience with independent schools is that they are more flexible with the curriculum and the ability group within the class. The curriculum is innovative, in places, but also very traditional and time is not spent changing the curriculum because a new government idea has been introduced. There is competition on the sports field and in the classroom. In DS's reception class he was doing Year 3 spellings with a small group and another group was learning their ABC. In the state school DS2 had to learn his ABC eventhough he was reading Level 14 in the Oxford Reading Tree books. This is not just my experience and I know plenty of parents whose children are in the same situation. IMO some teachers do not think outside the box and stick rigidly to the curriculum they have been given.
I also believe that independence is taught from an earlier age at independent school and there are many opportunites where children are expected to venture out of their comfort zone (not in a harsh way) on a daily basis.
But again all schools are different and this has been my experience.
Will your DC have a better chance at getting into GS from an independent ? Only if the testing is English and Maths and not VR/NVR. If you choose a school that is a large feeder to GS then it may well increase your chances but if you want an independent education it should be for much more than the academic results. It should be about the confidence, discipline, self-esteem and promoting the individual that most of these schools provide. I'd rather my sons could look an adult in the eye and have a respectful and eloquent conversation than have a clutch of A'levels, if I had to choose.