I totally agree.
I've worked mainly in state schools, but had a stint in an independent boys' school, and it was an eye-opener.
Of course the public school had a wealthy (mostly - there were a lot of busary students) intake, but there are still things the state system could learn which don't cost money.
One obvious thing is the huge emphasis on courtese, manners, behaviour etc. Having come from a tough comp, I was literally rendered speechless when, on my first day at the public school, a lad leapt forward to open a door for me. This was not what I'd been used to at the other schools. It costs nothing and would stand all the pupils in good stead for the rest of their lives.
Also - encouraging confidence through dealing with adults more on a trusted, equal basis. Students working with teachers closely to organise activities and clubs. More emphasis on public speaking - not long speeches or debates all the time, but just making announcements or giving a vote of thanks.
These things cost nothing, but would hugely raise self-esteem in state-school students for their adult life and prepare them for the world of work.