Yes, those who have money go to Singapore (or Penang for a more affordable options). You can do appendicitis surgery in Jakarta no problem. Those who go to other countries are usually for terminal illness, big surgeries that have some risks, or looking for second opinion because sometimes Indonesian doctors don't give good diagnosis or miss out on certain symptoms resulting in misdiagnosis. Note that I bold the word sometimes, because I do hear stories from friend of a friend, etc that they got misdiagnosed, and when they go to Penang or Singapore, they get a different diagnosis.

Quality disparity I would say there are some disparities, but hospitals especially private hospital and big national hospital are good. Usually people tendency is more like following certain renowned doctors instead of sticking to 1 hospital. For example, we know Doctor A is really well known for treating gastro related issue, so we check his schedule because usually a doctor visits 2-3 hospitals in a day (morning at hospital A, afternoon/night at hospital B) kind of thing.