I'm not quite sure why private automatically means better? I was privately educated in a good day school as were my siblings and you pay for smaller class sizes and parents (and most children, not all) being invested in their child's education.
But, teaching was not always great. In some subjects I felt there was a lack of ability to teach to lower ability students (I.e science for those who did not intend to do medicine/ dentistry at university) and I had private tuition for languages in addition to school fees! (GCSE level).
School trips were good, skiing, choir trips to Europe, day trips to Ypres etc. but nothing a parent couldn't replicate or outside interests couldn't assist with.
Yes, I have good grades, a good degree, a professional qualification. I expect a dedicated student (which I was) could have attained these at the local comp. friends certainly did!
You could definitely tell the kids whose parents scraped by. I was friends with some, didn't bother me but it may well have bothered the kids.
Our joint income does not extend to private (unless I go back to my profession which I didn't enjoy!) so I don't have that dilemma. We have done it the other way, moved to an area with good (some very good) state schools. Would that not be a better option?