In my experience, its not so much what the NHS won't fund but getting access to what it does fund in a timely fashion.
Of course there is dentistry, which is a complete nightmare... people trying to do their own extractions with a pair of pliers is the sort of thing you used to associated with the US but I've actually met some people who have tried that due to how poor NHS dental services are and how expensive private treatments are.
Weirdly enough the US is actually amazing for dentistry.
With any insurance you get 2 cleanings every year fully included and most routine fillings are almost completely covered. Easy to find appointments everywhere
My experience in Europe has been that it is super difficult to schedule anything. Waiting time of multiple months for new patients.
Yes, my (private) dental insurance in the UK is similar. If you go private then no real problem getting treatment as soon as you need it (I got an appointment in the same day recently when I broke part of a tooth).
Of course there is dentistry, which is a complete nightmare... people trying to do their own extractions with a pair of pliers is the sort of thing you used to associated with the US but I've actually met some people who have tried that due to how poor NHS dental services are and how expensive private treatments are.