Obviously, children learning in independent schools were going to do better under Covid and teacher assessments because the standard of online lesson delivery, during Covid was probably much higher and consistent. If parents are paying for education, they expect a full 9 am - 4 pm day of well-structured lessons, or could threaten to withhold fees, and probably look in on some lessons, just to check they are happy with the quality of the lessons and getting their money's worth. I should imagine that parents who send their kids to private school, are, in the main, more confident of themselves, when critiquing and assessing the education being offered, because highly-paid jobs, which supply the dosh for school fees, tend to require degrees, MAs and high level of academic training to get into, ergo private school punters are confident about demanding homework/ additional resources and individual support. That is the case for the small handful of independent students that I know.
Also, parents of kids in independent schools appear to have to deliver much of the explaining and supporting themselves, even in non-Covid times, from what I've seen among family members. Private school kids seem to get pages and pages of homework and holiday work which they are expected to complete, and parents expected to support their kids to finish, with a lot of pressure for this to happen. I expect the parents may have had to act as educators, also, during lockdown.
Some kids learning in the state sector may not have had access to computers, and, generally, some parents may have been less able to create a peaceful and supportive environment, if resources are stretched.