Sure, Chomsky's work doesn't have practical applications. Most scientific work doesn't. It's just that, for obvious reasons, you tend to hear more about the work that does. You mention number theory. Number theory had existed for a lot longer than Chomskyan linguistics has now when Hardy chose it as an example of a field with no practical applications.
> seems to reject the idea of falsifiable predictions entirely,
As a former syntactician who's constructed lots of theories that turned out to be false, I can't really relate to this one. If you look through the generative linguistics literature you can find innumerable instances of promising ideas rejected on empirical grounds. Chomsky himself has revised or rejected his earlier work many times. A concrete example would be the theory of parasitic gaps presented in Concepts and Consequences (quickly falsified by the observation that parasitic gap dependencies are subject to island constraints).
The irony here is that generative syntax is actually a field with a brutal peer review culture and extremely high standards of publication. Actual syntax papers are full of detailed empirical argumentation. Here is one relatively short and accessible example chosen at random: http://www.skase.sk/Volumes/JTL03/04.pdf
>After dominating academic linguistics for fifty years, it has never accomplished anything considered difficult outside the newly-created field
What does this even mean? Has geology accomplished something considered difficult outside of geology? I don't really understand what standard you are trying to apply here.
