Yes, but they are inconvenient, you had to line up in the morning for an appointment when I lived in Beijing, and scalpers would commonly grab most of the tickets early to sell to you for a few hundred RMB. I hear they have a better system now, something based on WeChat without the scalping.

Private hospitals were mostly staffed by doctors from Eastern Europe and Iran, at least the one I went to (UFH in Lido, Beijing), lots of rich Chinese customers. It was comfortable and you just made an appointment on the phone. You got to wait in a nice waiting room with a plush couch rather than a concrete brutaleski waiting room with plastic chairs, private birthing rooms for babies rather than a community birthing room.