This is exactly it. For things like buying a house, it now doesn’t matter if you do brilliantly at school and university and get a well paying job, because the person with the inheritance will always outbid you.
Or even if you can buy the house, you’ll have an enormous mortgage and they won’t. So they can do things like send their kids to private school and pay for extra tutoring or whatever which then perpetuates the inequality. And of course they can hand over a bigger inheritance to their kids.
As a pp says, social mobility has rolled right back - we are back in the pre war era in social mobility/inequality terms and are headed towards Victorian levels.
Personally I think the govt should introduce CGT on gains on homes, but do it gradually and with warning. You’d get a lot of house sales in the short term trying to beat the incoming taxes. And then long term with CGT on gains on your residence (like on other investments) it would no longer be seen so much as a piggy bank but as a home. And of course, it would raise a whole load of money to prop up the various state systems that help reduce inequality (like decent state health and education provision).