OK then.
Re the OP's query re whether private school teachers are more highly qualified, in this thread many of those with kids at private schools have claimed this is the case, which may or may not be true - what is more significant is their unquestioned assumption that having more highly-educated teachers inevitably leads to pupils with better results. I'm not aware of any research that proves that assumption is fact.
What the vast body of research on educational attainment has repeatdly shown, though, is (a) that children with more highly educated mothers (not fathers, interestingly) do better academically, and (b) that the surest and quickest way for any child to make progess is one-to-one tuition.
This ties in with the discussion as it appears to be the educational level of the parents rather than teachers we should be focusing on - if we want to make the biggest impact. But a bit unhelpful for hose from less well-educated backgrounds!
It also suggests, though, that if you are a well-educated mum yourself the best way you can help your kids - contrary to Xenia's suggestion - is to devote some of your time to helping your kids 1-1. (Rather than working fantasticaly long hours to pay to put them in a smaller class size of say 15.)
If you can't (not well-educated) or won't give your own dcs 1:1 attention, then rich parents often resort to the second-best of paying tutors or going for private schools, which due to the smaller teacher:pupil ratios, NOT the 'better' teachers, can go some way towards catching up with those pupils lucky enough to get 1:1 stimulation at home.
This musing came from contemplating a large nuber of educated and intelliget friends and acquaintances whose dcs did not achieve as highly, academically, as you might have expected based on their genes. Largely they had parents devoted to careers and hence educational support was generally farmed out to paid providers/schools. Obviously, my cross-section may be atypical and I don't wish to knock full-time working parents, many of whom have no other choice. But there does seem to be a correlation between v v long working hours and kids performing less well.
Xenia? Over to you.