It's more subtle than private vs state or middle class vs whatever.
My DD is at a private school which provided amazing lockdown education, which meant that the kids got on with their school day needing very little support from parents. Lucky kids (and parents! )
If the school hadn't done that (and not all private schools did) it would have been pretty unequal within the class. A lot of DD's school friends had both parents working long hours during lockdown: many doctors on the covid front line (sometimes both parents), or in key roles in national broadband infrastructure, or just dealing with the crazy which covid brought. They weren't available to homeschool their kids.
Other kids (at both state and private schools) have a SAH parent, or had a parent on furlough. One-to-one teaching for 6 months is arguably also a pretty good educational advantage. Lucky kids.
Some kids had neither school provision nor a parent able to help over lockdown (even if they really wanted to). And of course there were children without the equipment they needed, or in overcrowded accommodation, or stressed/struggling families.
So whilst there are some links between parental occupation/income and children's experiences, it certainly isn't as simple as state/private. It's a whole host of factors which aligned to give kids very different experiences of lockdown.