I’m not really knowledgeable about much of German history, but I do know it is misleading to talk of “German history” as if there was one nation-state called “Germany” (until recently.)

I do know that immigrants to the US during those years (after the US Civil War) were often fleeing poverty more than anything else. in previous centuries there were people fleeing Germanic speaking states for religious reasons (see the history of Pennsylvania, for example.)

Papa John Papa claims his ancestors were fleeing in 1867 because of some onerous government growing too large.

Well, not being the expert I am, checking on Wikipedia for that period of what was happening in the German states, here is what I found:

During the roughly three years of the North German Confederation its major action existed in legislation unifying Northern Germany. The Reichstag decided on laws concerning (e.g.):

free movement of the citizens within the territory of the Confederation (1867)

a common postal system (1867/1868)

common passports (1867)

equal rights for the different denominations (1869)

unified measures and weights (with the obligatory introduction of the metric system)

penal code (1870)

I suspect that Papa John’s Papa’s story is just more made-up family mythology.