Even when you have an NHS dentist , like I do, they keep kicking us off the appointment list as the dentist leaves for a better paying job - A proper mess.
Dentistry was removed from NHS services many decades ago. Dentists are self employed and can have a contract with the NHS, if they want one. Often a lone dentist will hire a chair in a dentist practice. Not everything is paid for by the NHS contract. eg. there are no white fillings at the back; silver crowns are used at the back; even front crowns are basic as are dentures; some treatment is not not covered by the NHS contract at all. Some NHS dentists will not clean and you pay extra for that with somebody else at the practice.
The same with GPs, they are self employed too and have a contract with the NHS. Some are partners of a limited company and others are hired by that company. They can choose to keep taking people on their books, even if they can't cope with the numbers they already have.
Trusts are NHS hospitals. These vary from area to area and from country to country. You can have private treatment at an NHS hospital and that Trust has that money.
Some people decide to have a private dentist and GP. These are not on NHS contracts; but some are and see private patients (quicker) too.e.g. when I needed an MRI, I went from the Consultant's office at the hospital, to the front of the queue for the scanner, with a note from the Consultant.
Some people decide to pay privately in other countries, for dental work and medical treatment.
This is nothing new. The NHS doesn't cover everything and never has.
I paid privately for my children as they got to see the Consultant straight away and didn't have to wait, or have to see somebody who is training. As adults now, they have continued to use private GPs (and dentists), and for their children too.
Many Brits report that treatment abroad is cheaper than private in the UK. Others live in some EU countries where residents pay, but say that in that EU country it is good treatment and cheap.