Non-emergency care at Hospital Aleman can be tiring. And difficult to get an appointment. I usually don't have a problem with that - I don't like to go to the doctor anyway and making an appointment for something non-urgent doesn't bother me to wait. But like RichardRPTownley, the couple of times we've had urgent requirements (and once a full-blown emergency), we were taken care of right away.
Even with a plan (which I have) you can wait weeks for a turno, depending what it is you need. It also depends on how you go about it. HA has its central facility, but if you are a member, there are also a lot of doctors associated with the hospital, all over the city and province, particularly for specialities, that you can make appointments with on the website (or from the registry of doctors/clinics) and usually get an appointment even the next day.
Our youngest was interned in HA for two days last year. She had a private room, very clean and nice, and even had decent food, nurses came by every so often to check on her and make sure everything was good - it felt like a safe place. Since she was a minor, by law a relative had to stay with her.
But what I've seen in the public hostpitals, if you have to stay, it won't be nearly as nice. And you have to have someone stay with you, at least for part of the time, as there is no help to take care of you beyond whatever the doctor may order for medication (including food, clean sheets, etc).
BTW - having a chronic or developing issue is a lot different than an emergency. HA sees a lot of people both members and non-members. While I remember times in the past that HA was much more responsive on ordinary things, I have to say that I am very happy with how they take care of us when we've had real problems.
And I would submit that many people who can afford a visit but don't have a plan would prefer to pay the money than go to most public hostpitals here. Not just expats
Maybe my inlaws don't go to the correct public hospitals, but the stories they tell me make me want to stay far away from them. HA is more resonably priced than Swiss Medical and OSDE as well, which is a reason it's popular with many expats and middle class folk.