The reality is that yes, if you have the money, you can pay for a better standard of care, and part of that is convenience.
So you can go private for dental, you can go private for medical care, say if you have cancer, and bypass the waiting lists.
It may not be 'fair'. But it is reality, and pretty much always has been.
It's not allocating appointments on the basis of wealth, it's allocating appointments on the basis of what the client is prepared to pay. Some of those clients will not be wealthy - they'll be desperate, and will prioritise their dental care, and go without in other areas to afford it.
Dentists, just like any other service provider, are entitled to seek to maximise their profits.
Where I come from, all dental care once you leave school, is private. There is no public provision.