Probably not one that's very useful. If you think of it as an artistically-related metatheory construction kit (so, a meta-meta-theory), that would probably describe what the bits that aren't shit are useful for. Gwern (per comment parallel to yours) probably wouldn't benefit, having deep cultural knowledge in at least one area, though he might find useful bits to avoid having to reinvent terminology.
(Oh, and it's not one thing - any area that has something that corresponds to a "modernist" outlook, where progress can occur, is likely to spawn a "postmodernism" which involves questioning every assumption including that the right questions are being asked. Remember that it sprang up after World War II, which was seen by the postmodernists as the horrific reductio ad absurdum failure of several decades-long modernist programmes.)
What I mean is that dismissing this icky squishy cultural "what is meaning?" stuff as a diseased discipline does not somehow mean you, as a bounded human intellect in the world, are immune to the problems the toolkit-constructing toolkit in question can somewhat alleviate. Because the proper study of mankind is man. Squishy and infuriating as humanity is, directed cyclic graphs of preferences and all.
FWIW, don't read original Derrida unless the wall opposite needs a few more dents in it - stick to commentaries.
I challenge anybody to come up with a higher-density description of postmodernism than 'Fallacy Of Gray Plus Meta Everything with Leftism'. Remember that it arose out of an enviroment that was far too absolutist and ignored meta-analysis.
Related to: Privileging the Hypothesis
-- Paul Graham
-- Doug Henwood
-- Eliezer Yudkowsky
Here are some political questions that seem to commonly get discussed in US media: should gay marriage be legal? Should Congress pass stricter gun control laws? Should immigration policy be tightened or relaxed?
These are all examples of what I'll call privileged questions (if there's an existing term for this, let me know): questions that someone has unjustifiably brought to your attention in the same way that a privileged hypothesis unjustifiably gets brought to your attention. The questions above are probably not the most important questions we could be answering right now, even in politics (I'd guess that the economy is more important). Outside of politics, many LWers probably think "what can we do about existential risks?" is one of the most important questions to answer, or possibly "how do we optimize charity?"
Why has the media privileged these questions? I'd guess that the media is incentivized to ask whatever questions will get them the most views. That's a very different goal from asking the most important questions, and is one reason to stop paying attention to the media.
The problem with privileged questions is that you only have so much attention to spare. Attention paid to a question that has been privileged funges against attention you could be paying to better questions. Even worse, it may not feel from the inside like anything is wrong: you can apply all of the epistemic rationality in the world to answering a question like "should Congress pass stricter gun control laws?" and never once ask yourself where that question came from and whether there are better questions you could be answering instead.
I suspect this is a problem in academia too. Richard Hamming once gave a talk in which he related the following story:
Academics answer questions that have been privileged in various ways: perhaps the questions their advisor was interested in, or the questions they'll most easily be able to publish papers on. Neither of these are necessarily well-correlated with the most important questions.
So far I've found one tool that helps combat the worst privileged questions, which is to ask the following counter-question:
What do I plan on doing with an answer to this question?
With the worst privileged questions I frequently find that the answer is "nothing," sometimes with the follow-up answer "signaling?" That's a bad sign. (Edit: but "nothing" is different from "I'm just curious," say in the context of an interesting mathematical or scientific question that isn't motivated by a practical concern. Intellectual curiosity can be a useful heuristic.)
(I've also found the above counter-question generally useful for dealing with questions. For example, it's one way to notice when a question should be dissolved, and asked of someone else it's one way to help both of you clarify what they actually want to know.)