It's still a big challenge to find one in many areas (like mine). You call, get on the list, and if it's not urgent it will take weeks or months, you might have to drive quite a bit to get where they send you, and don't expect to find people happy to see you when you get there.
Full as in "Yes, we're accepting patients, but the waiting list is two years, and we won't see you for an emergency until then."
I had a lot of NHS work done last year. I didn't pay much because I was used as a guinea pig by a couple of freshly qualified dentists who wanted to improve their skills before being promoted to private practice.
One dentist was good, one was kind of okay, I guess.
The deal wasn't explained. It was "And this is the NHS dentist you'll be seeing." Which is maybe not as ethical as it could be.
I've paid for private care in the past and it's never been significantly better. You get more choice of filling/crown material, but the actual work seems to be as good - or bad - as NHS work.
By far the best work was done for free by supervised students at an NHS teaching hospital in the 80s. Most of it has lasted for decades.
The issue with dentistry is possibly that there's no distinction between primary health care - the "have this tooth out or die of sepsis" kind - and purely cosmetic surgery, which is strictly optional.
There's also some overlap with products like implants, which improve looks but can also work better than fillings/crowns.
I can't think of any other branch of medicine which has the same tension between health and appearance.