I think it is more helpful to look at net income in doing the sums as to what it is affordable. A gross salary of £200K is actually about £120k net, as some of it will be taxed at 50%. Given the amount of your mortgage costs, you will not be able to afford to put 4 DC through private schools, particularly as they will inevitably overlap. Assuming £20k per DC per year for a good private day school in London or the South East, that is £80k out of taxed income.
Also, please bear in mind that law, in common with other professions, is not the safe option it used to be and there are no guarantees that your DH will maintain his current level of earnings.
If private education is important to you both - and it does sound as though DH is making the decision alone - then why don't you raise the funds by re-mortgaging your house, as it appears to have substantial equity. There are some good deals around at the moment.
Only you can decide what is better for your DC and what is right for one may not be right for all. There are some excellent state schools and some rubbish state schools, ditto re private. The received wisdom - with which I probably agree - is that a bright DC will probably do well anywhere with good parental support - but those who struggle or who have special needs will almost always be better off going private. A private school is able to offer smaller class sizes, more accountability to parents, and a wider curriculum: in particular, the opportunity to study the Classics - Greek and Latin - which is sadly lacking in most state schools (well done Boris for your efforts in bringing it back!).
I am not sure that I agree that private school pupils necessarily have more confidence. Many do acquire a veneer of polish - a way of speaking an acting - which appear confident. However, they have the same insecurities as everyone else, possibly more so as they have only mixed with people from a narrow social spectrum and can easily find themselves outside their comfort zone.
I would worry about your DH's belief that he can buy the DC places at Oxbridge. If anything, it is probably 'easier' to obtain a place from a state school, partly because of the pressure on the best universities to admit more state school entrants but, also, because the universities perceive that an applicant with A* grades from a state school is likely to be a better candidate than one with similar grades from a private school.
Your DH's attitude will also put great pressure on your DC to succeed academically. You need to accept that they are individuals who will ultimately need to choose their own path. I don't want to worry you but one of my DC's contemporaries dropped out of Oxford after one year to run a stall at Portobello Market because he was disillusioned with academia: another turned down a place at Cambridge to study hairdressing to 'spite her parents' whom she felt were putting too much pressure on her!
Good luck, whatever you decide!