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I don't know the Frankfurt school scene, but can add some generic remarks (most of which have been discussed in detail in other threads, which you should read!):

• Private schools in Germany are expensive (at least 1000€/month per child) and not necessarily better than public ones.
• Bilingual schools are geared towards German students learning more English, not foreigners learning German. This might now be what your kids need.
• Public schools are genarally good (and of very similar standard,so there is no need to shop around) and free of charge, but all teaching is in German. Some offer "integration classes" to bring foreign kids up to speed (not only in the language).

Nearly all public schools are fine. In many countries the public schools tend to be bad and the only way to get a good education is at private school. This is not true in Germany. One doesn't really get a choice though. One goes to the one assigned to you, usually the nearest one that has space. For children with special needs or say they don't speak German yet and need remedial help with the language then it is possible a school a bit farther away might be decided to be the one best suited.

Most private schools and certainly international ones tend to be very expensive and not necessarily any better academically. The influence one can have on their kids’ school is their choice of where to live. If one lives in the poorest part of a city, rather than one of the richer areas, then your kids will also go to school with kids of poorer families. Such families might tend to be more blue collar workers than academics. Thus the attitude towards education can be different. Not that there is anything wrong with being a laborer per se.