Is "swimming in broth" a common cooking technique for pasta and broccoli rabe?
I am not sure whether it is a common way to prepare it in the US, but I would say that it is definitely not the traditional Italian way to serve pasta salsiccia e friarielli which is instead a fairly dry pasta dish.

Of course nothing stops you from making variations on the theme. For instance, I know some people who purée the broccoli rabe.

However, in Napoli (where the dish is from) you would generally have it dry.