Technically, we only have private-sector GPs, but most work mainly under contract to the NHS. This is a consequence of how the NHS was created in the 1940s.

Some offer private services too and some only do private work, but try it for yourself. Throw a pin in a map of England and try to make a private primary care appointment. You'll often end up in the nearest city or large town, maybe 30-50 miles away if not on the big city spine. Not convenient, and then there's the cost, often £150-200 for a first short appointment urgent package. Unless you're already subscribed to private healthcare at "from £11.32 to £127.89 per month" to quote one private mutual, it's not an option for most people (and why should it be needed if we've paid our National Insurance...)

No, not really. And even if there are, you're not always guaranteed you can even get onboarded into a local GP surgery. So most people are stuck with whoever they are currently with.