Anecdotally, in the UK this now normal. Apparently you have to beat the system by calling at a certain time and saying your problem is urgent. Obviously, all the people you are referring to know all about this; it's just suckers like me who have to suffer.
The NHS is free at the point of use in the UK - except for dentistry which is only free if you have a low income, etc.
Most folks (in my experience) go full on private and pay a monthly fee plus insurance. It isn't expensive (unless you need some major work doing) and the standard of dental care has improved many fold as a consequence.