The whole topic deserves an entire book. Even then, the answer "who is the most meritocratic" won't receive a straight answer. The point that British culture does not allow the prime minister to send their children to fancy schools is a meaningful one to make, among many others.
It puts the UK closer to the Scandinavian/Northern European countries and the USA closer to less-than-democratic countries (MENA, Africa, Russia, China) on this particular axis.
To be clear - I don’t care where David Cameron, Theresa May, Boris et al send their kids to school. I care that such schools exist effectively to officially stamp you as a member of the upper class.
Edit: I hope this doesn’t sound like I’m annoyed with you or the original commenter! I have some strong feelings about this but I am aware that you didn’t create this system, that you are not standing up for it or trying to protect it. We’re just having a discussion and as you say it really deserves an entire book
Whoever wrote this is being a little disingenuous. The pipeline from a handful of private schools to the upper echelons of power (via Oxbridge) in the UK is very well known. The fact that the press wouldn't let the PM away with sending their kids to Eton or Harrow doesn't really change this or make the country more meritocratic, it just means we all pretend they're just like the rest of us for a few years.