On topic:
My baby girl is 6 weeks and I've have taken her to a public clinic, twice

. Public clinics aren't that bad. If you plan correctly and choose a decent clinic, then you should not have any issues.
I've called around (Dischem, MKem, Clicks, BreastFeedingClinic in Panorama, etc.) and I had no luck finding a private clinic who offers the 2 previously mentioned vaccines for free. I had to pay R900-R1200 at pharmacy clinics who were also understaffed and with a long list of patients. Eventually I opted for going to a gov clinic.
I chose the Goodwood clinic since it was the closest and I could get there early. To my surprise, they were cleaner and more organized than I expected.
But instead of sitting there with my baby for an hour until she is administered, I got there early in the morning (when the doors open @ 7am) and handed in the clinic card so that I'm 1st in line. Then I went home, fed her and eventually prepped her in the car seat. I got there exactly 8:45am just in time for the nurse to administer the vaccines. Total time in the clinic: 20min
Sometimes all it takes is a bit of preparation. I honestly still prefer the convenience of having cleaner and private consultations, but using a gov clinic ain't that bad as one would think.