I wanted to get a sleep study done to get to the bottom of my occasional insomnia. My PCP at the time put in the pulmonologist referral (because this hospital network required a referral to get a sleep study done, which is insane in its own right, but that's neither here nor there).
This was in January. His next available appointment? March! For a friggin sleep study! And I have a HDHP + HSA and was going to pay full boat!
(He also prescribed me with an antipsychotic that can double as a sleep aid at low dosages, which I didn't take because five minutes of research into it made it obvious that taking that for extended periods of time is bad news bears. I stopped seeing him after that.)
(For those looking for answers like I was, I eventually stumbled upon Cognitive Behavioral Therapy for Insomnia, or CBT-I. It's essentially a rigid sliding window-style sleep schedule plus challenge therapy to make you rethink how you approach sleep. I've been doing this for six months now and it has been transformational. Best part? No meds!)
Getting a routine gynecological appointment for my wife any time in the next six months is impossible. I gave up on trying to get an appointment with an allergist for myself, even after getting a referral. What you describe is common in the US too, even for those with means. I cannot imagine what it is like for somebody who doesn't have money and insurance.