@dalrympy, agree with this completely BUT it does depend on what state provision is nearby.
Op, it really depends on the private school. I have 2 nearby, on paper they should be similar but ethos is very different. One has a high academic threshold for entry and as a result it is a pushy school, with pushy parents. Good results, great facilities though and there is a positive image associated with going to the school.
The other has relaxed entry and the school is much more about the whole child.
I have dc at private and was recently commenting to a parent who was debating choices. I said that I knew, logically, that the school wasn't perfect but I struggled to complain about any aspect. Not all teachers are brilliant (many are!) but any bad teacher doesn't last long.
Bullying is definitely stamped on. They are highly responsive to parents but the parents on the whole are not demanding. There is a group WhatsApp but it's helpful, such as sharing info, rather than critical of the school. I guess the very big downside is £££. When it comes to paying increasing fees it does make me think of what else I could do with the money..but equally I wouldn't take my dc out of school.
My dc went to state primary and had a fabulous time but I know 2 parents who have removed from state at primary due to poor teaching (lockdown) and another for bullying. I think funding by government is an issue unfortunately. However dc can easily recover once they go to secondary school.
If you are in a debate with your DH why not state for infant or primary and definitely do private for secondary.