OP:
I've heard similar stories to yours from more than one person. A parent lives near good state schools, has no obvious reason to go private, but "I've done the sums, we COULD afford it if we tried, and I went to a private school myself so it seems normal to go to a private school. Aren't I short-changing my kids if I don't pass on to them the private education that I had? I feel a bit guilty, thinking about it."
I think it's normal and natural to feel that way.
But.... a generation ago, middle class parents could send their kids to private schools without any hideous financial deprivation. Since then, it is a different world. The fees have gone up and up and up in real terms, and they will continue going up over the many years until your third child is finally out of school (and then you've got three sets of uni fees to pay). Have you factored in all the cost increases that will inevitably happen over those years?
Meanwhile, most other professional parents have given up on the idea of private school too, meaning that the state school system is now full of parents with great careers (plenty of "connections" there...) who take their kids' education seriously. As someone else said, look at the top 20% achievers at your local state school and then compare it to the private schools. Is there much of a difference?
I am not going to deny that private schools are on average somewhat more academic and push the kids a bit harder--smaller class sizes and better behavior too, in most cases. They also tend to have nicer sports and music facilities. But there are almost certainly cheaper ways to plug the gap. Have an educational environment at home, do educational things at weekends, pay for good extra curricular and holiday clubs/camps, invest in tutoring as and when needed. The amount of "extra individual attention" that a child gets in a class of 20 as opposed to a class of 30 is not much. If I want my child to get individual one-on-one attention, I think it make a lot more sense to pay for targeted individual tutoring here and there in particular subjects over the years.
At the end of the day, your kids may thank you a lot more if you have some money set aside to give them for a deposit for a property, considering the state of the housing market in the UK.
I'm absolutely not snarking on the idea of private education: it really does makes sense for some people. If your child has special needs, they may benefit from a particular type of school; some children have specific talents that may make a specialist school worthwhile; some people are tied to areas where the state schools are really poor; some people just have money to burn and appreciate the "reassurance" of knowing that their child's school is unlikely to fall below a certain standard; some couples both have high-powered careers and want a school that provides schooling, homework supervision, tutoring, extracurriculars within a wrap-around package, because they don't have time to supervise their child's education and homework and cart them around to ballet and swimming. Horses for course.
For most middle-class familiesesp if one partner only works part-time and there are three kids and the local state schools are goodit is almost certainly not worth it.