no

UK "public schools" are pretty much a subset of private schools:

https://en.wikipedia.org/wiki/Public_school_(United_Kingdom)

In the UK, public & private are pretty much synonyms in the schooling context, traditionally the public schools were the "grand institutions", Eton, Harrow etc, but you'd struggle to find anyone who would make that distinction in the present day. If you did, they probably went to one of them.

I must admit that I'm in Scotland and my son went to a couple of private schools (though they seem to call themselves "independent" schools) - never heard anyone referring to a public school other that in UK media discussing Eton etc.

I can well believe that the madness of "public school" being a private school is a peculiarly English phenomena.

For clarity:

Public schools - a handful of top level very expensive private schools, usually have existed for centuries, the children of the highest levels of society tend to go there, a great foundation for a career in public life. Examples include Eton, Harrow, Winchester, Marlborough.

Private schools - the rest of the private schools, less expensive, but less advantageous.

State schools - free, provided by the state for children in a local catchment area, this is where the rest of us go. Most are run by the local government, but there has been some experiments with 'foundation schools' which are tax payer funded but privately run.

A good breakdown.

There are also a handful of true grammar schools left. These are free to attend, but are selective based on ability. There is debate about the extent to which raw ability is really being tested, and not preparedness for the test.

There are also private schools that call themselves grammar schools. These were once free and selective, but converted to fee-paying at some point. They like the prestige of being called a "grammar" but don't be fooled!

The lack of true grammar schools has been a big hit to social mobility in the UK. Now, getting into a grammar requires not just passing a test, but also having parents rich enough to buy a house in the catchment area. Public school alumni dominate top industries and government.

The UK "free school" approach has been a mixed bag. On the one hand, various academy chains have managed to turn around failing schools. On the other, some parents are left with no choice but to send their child to a religious school due to catchment areas and waiting lists.

As a tax payer, I don't want my contributions going towards religious institutions.

The loss of "special schools" for students with severe special needs has not helped anyone (imo).

A lot of religious non-selective schools tend to get better results than non-selective secular schools.

You see a few parents suddenly get religion as their child turns 9 or 10 and they realise they are unlikely to get into a grammar school.

... or decide a buy a second (or third) house conveniently located in the good school's catchment area ...

And funnily enough a lot of the people that do that "don't believe in private schools".

State Schools are also divided (in some parts of the country, such as Kent) into selective and non-selective at the age of 11.

If you pass a specific test (the 11+) you can go to a selective grammar school.

There also used to be government funded scholarships to private schools for the very brightest underprivileged children but that was stopped by Labour, the "party of the Working Class".

How do you get your child into one of those public shcool? Do you need money, or connections, or does the child need be talented? (The Wikipedia article didn't tell.)

In general, lots of money. Most do scholarships, so if your kid is really bright you might get one of those (usually reduced rather than no fees), they do this to bump up their examination results. Connections seem less important now than it used to be, but many will give preference to those whose parents or siblings attended.

Officially money, an interview and an entrance exam.

That's not to say that all things being equal you wouldn't stand slightly more chance if you were a member of the aristocracy.

Money. If you can pay, exceptions will be made for everything else.

It helps if you can trace your ancestors back to the Normal invasion (the winning side, obviously). I'm only half joking.