I would send them to a state primary, and then reconsider again at secondary age. I honestly believe that for primary school, parents make most of the difference - if you can take your kids to museums, galleries, historic sites, talks, science events and do arts, crafts and stuff at home, they’ll succeed and have a great time. Kids who go to a private or grammar school from a state school are likely indistinguishable from private prep school kids with the exception of better sports provision at prep schools, sometimes better MFL (although usually the U.K. is bad at both regardless of school...) and music. However those can be provided by a parent, especially if they’ve got more money to do so.
If when they are coming up to secondary school age, they want to go to the private or grammar, I would pay for tuition for their entrance exams. At that point you can make the choice - they may well not choose private or grammar, or might prefer one or the other.
A private school may be massively beneficial for a child with great musical talent, for example, or a child who might benefit from the focus of small classes, whereas another might not want too academic an environment like a grammar and could do better in larger class sizes. And again, it can change for the 13+.
Money you don’t spend on private school doesn’t have to go straight to the house, but could allow you to pay for summer schools, MFL tuition (or exchanges abroad), revision courses for school holidays, memberships to societies, sports equipment, music lessons, educational holidays, trips to different historic sites etc. I would prefer to keep the money, put it towards the house but for smaller than is exactly necessary so that there’s money for special extras such as that.
With parental support + supplementary and fun education, theh will almost certainly succeed and do brilliantly, I know very few kids who haven’t succeeded in such circumstances. However I know many people who have succeeded academically and with many other talents and interests who have not got financial stability.
Financial stability/security allows for them to focus on developing their career and themselves as a person - to take a less well paying internship because they know it opens more avenues in the future, to go for a masters degree, to start a family at a younger age, to volunteer more because they don’t have to work as much (or at all) at university and in the summer holidays.
It does depend on the state school though. Our catchment school has ‘well below average’ progress 8 scores, 21% Grade 5 or above at GCSE English and Maths. I would send my child to private instead. But the eldest will be going to an Outstanding state primary and we would likely be able to get a place at a different secondary, the two nearest who usually have out of catchment places have average or above average progress 8, GCSE results etc and I would send them there as I believe the environment would allow for them to succeed academically even if they’d be pushed less.