As you can see, there's no easy answer.
Independents are not highly selective at 11+ in many areas outside of the south-east and big cities - they have entrance exams, but are not hugely over subscribed, so most applying will get a place. It the south east and London, there are a mixture of 11+ schools which are highly competitive and again those with an entrance exam, but which most kids who apply to get into. So, when you speak about independent secondary, there is a big difference between considering a highly selective one (will need to be bright and probably tutored or home-prepared for exams) and one that pretty much anyone who will pay the fees can go to.
So....if state options are equally disappointing at all levels, personally I would pay for secondary - either to 16 or 18 depending on funds. In most areas of the country, your kids won't have to be super bright to get a place.....in the more competitive schools in Lindon and the South East and some others dotted around the country, they will have to be and so 'top-ups' to a poor state education might be needed.
Answer to the question depends on the state provision in your area at each age point, the Independnet provision at each age point in your area (lots of areas outside of the south east have surprisingly few independent schools and often those that don't seem to access the academic heights, even if they do provide good value-added and a stable, secure education.....but be careful,mas there are lots of 3rd rate independents about too which might not be worth paying for) and also your individual child. What suits and is best for a very bright, out-going self-starter might be very different from what suits an average shy child, or a child with SEN, or the many other child types which exist)
If all other things are equal and one phase to be chosen, I'd go for secondary though.....but only if the fee paying options offer significantly more than the state options.
So your answer is to really research and know about the local state and independents at all age levels and to get a good understanding of your child and how to measure 'good' education, so you can see what is the best use of the money for your child, in your area. Gathering that info is a big research project, but one which is necessary and will yield different results for each family.