That is just wrong.

First understand that at that time there were more than 100 German states all with different laws etc. Poland was divided at least twice by the superpowers of Russia, Austria and Prussia. So in general the idea of one Germany or one Poland was foreign to that time.

The general image we Germans have with being efficient etc pp? This has a lot to do with Prussia and specially with the institutions that were comparatively very early very far advanced in Prussia: the military and education.

I am by no means an expert on the subject, but I know a few things through the history of my family. My ancestors were teachers for the kings of Prussia since at least ~1650, btw. the prince of Brandenburg would only get the title of a king (king in Prussia) in the year 1700 (or 1701 can't remember). However, we could make a living by maintaining a highschool in a Prussian village since those days.

What we can learn from that alone is, that even the rural population experienced a certain degree of literacy. At least enough to support a guy entertaining a classroom.

I also know that there was a law about compulsatory visit of highschool as early as 1717. (This also indicates a fundamental cultural difference between Prussa and the USA.)

I know that college students used to get free meals in the houses of the bourgoise families in exchange for teaching the children.

These students who moved from the rural sides to the big cities to study there, would later organize to bring national unity and democray to Germany. (As any friggin WW2 movie shows: it didn't pander out. Oh, well.)

In regard to "paid by rich families": there were no rich families in Prussia or Poland and that time. To be rich your either have to steal like a French nobleman or be a capitalist like a Dutch trader or an American printer. Being rich is a very relative term.