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The TLS guide to personal statements states that you can write a problem/solution style PS in which you discuss a problem you see with a legal system or society and what you would do/are doing to solve it (structuring your statement, point #3 http://www.top-law-schools.com/statement.html).

I've got some pretty strong feelings about the way Congress works, and I think I could convincingly argue for why it's a problem and the deleterious effects it has had on society. Would this make for a good PS, or would adcoms just wonder why I've written a political polemic that has nothing to do with law school?

It would REALLY depend on how you write it, and somewhat depend on the nature of the "strong feelings" and how repellent they are or aren't.

By way of illustration, my mother could completely honestly state "I've got some pretty strong feelings about the way Congress works, and I think I could convincingly argue for why it's a problem and the deleterious effects it has had on society." BUT, if she wrote an essay about it, it would come to light that her "strong feelings" are rooted in her Christian fundamentalism and her notion of a "problem" with Congress is that it's bound by a secular Constitution and not by God's law. Such an essay would probably not be an asset to a LS application.

If your statement was clearly and logically written and demonstrated an ability to think about things in a sophisticated manner, it could be good. If it would just be a two-page vomiting-up of your American Government and Politics 101 notes, reconsider.

I really don't think that the problem/solution approach is a good one. And saying you have a problem with Congress could easily elicit the following response from an adcomm:

"Really? A problem with Congress? That is really, really unique and profound.... NOT!"

It is possible to make this theme work, but you run the risk of coming across as highminded and idealistic.

I've got some pretty strong feelings about the way Congress works

don't we all? remember though that you are applying for something where a substantial number of applicants studied political science (which I'm guessing you did as well), so I would think (though I'm not sure) that that specific PS would scream "look! I'm like everyone else!"