This is something I don't understand of the US and other "developed" countries health systems.
Here in Mexico, i can get to doctoralia and make an appointment with a high specialist , like a Vascular Neurosurgeon. All by myself. The first visit will to maybe $200 usd (and that's very high) and normally it's a week away at most. No waiting on the private health option.
Of course if you go with public health, it's a disaster and a shame.
What other countries describe is the equivalent of your public health system. If you're willing to pay out of pocket, like you describe, you can have a similar experience in the US. However, that can get expensive quickly since you normally don't have just one appointment with a vascular neurosurgeon.
For most things, I can do the same in the US. It has a lot to do with your insurance provider. E.g. UnitedHealth is pretty well known for requiring pre-authorization for every little thing. But BCBS, which is who I have had most of my adult life, is far more relaxed about it.