But that's exactly what your source doesn't show. There are more Polish people living in Chicago than in Krakow; it's the largest population of ethnic Poles in the world anywhere outside of the largest metros in Poland itself. And they don't effectively exert power as a bloc. Your source shows one bloc, of Irish; today, the most effective wielders of power in Chicago are Black. There's no coherent immigration story to tell here.
It comes off a little bit like it would if you claimed that immigration brings with it organized crime, because La Cosa Nostra was dominated by Italians. But LCN is not in fact the story of Italians in America, and wasn't replicated by other ethnic blocs.
People share affinities and affinities structure interactions, and naturally some of those structural affinities are going to be ethnic. But if they weren't ethnic, they'd be religious, or political, or economic, which is what US history actually demonstrates.
If you're going to make the case that any of this matters in Chicago politics, though: cite the immigrant bloc that controls and distorts Chicago politics. Which ones are the illegitimate aldermen? I don't like most Chicago alderpeople, so you're not going to hurt my feelings.