This is the correct answer in a lot of cases and even if their hukou is in a modern place like Shanghai, the best hospitals and latest medical treatments are often not part of the basic medical coverage. Long lines and long waits happen for many services. A two tier system of public and private care exists with the rich able to pay for private doctor, private rooms, and advanced care. I lived in Shanghai for 11 years. Managed to have a heart attack there and got great emergency service at a fraction of the cost in the USA. After care was long lines and long waits at the local hospital so I went to the international clinic for that.

Would it also be fair to assume a rural province might not have the specialist you need?

I can only speculate, but I assume it has to do with the government structure. China's leadership is much more insulated (electorally) from the population than the European leadership. Said another way, European leadership is more accountable to their constituents and social safety nets are popular.

Chinese leadership are more accountable to the party which is just a subset of the population, not the population at large. I assume those in the party are on average better off and have less need for a social safety net, so they are less likely to demand that from the leadership. This is pure speculation though.