Don't agree on that. I'm not privately insured and never felt being not treated good enough by any doctor.
In fact although I am in compulsory health insurance (gesetzliche Krankenkasse) I have two doctors where I can call in so they put me on the list already, then tell me approx how long the waiting time would be, so I can leave later and don't have to sit in the waiting room for ages. Quite handy.
Since you were asking for a gynaecologist, I'm at
Praxisgemeinschaft Alsenstraße, they have a general practitioner, internists and gynaecologists. I only go to the gynaecologist, there's two of them. Dr. Möller is very nice and friendly, a very good doctor. Her English isn't great, but she tries. Dr. Bodelschwingh is the other one, only been in her practice once for an emergency, she was nice, too, think her English is quite good, when she noticed my fiancée is English she spoke English with us.
They also have a least two English speaking midwifes (in case you need that

)
They are in Hamburg Eimsbüttel, right at the border to Altona (or is it the other way round...?) anyway, it's close to S-Bahn Holstenstraße, a 3-4 min. walk from there.
About English speaking: having a quite young doctor does not mean they speak good English - my family doctor is way over 50 but speaks English quite well, whereas my dentist who is in his late 30s or early 40s really struggles and usually asks me for translation if he has to explain sth to my fiancée (who is also patient there).