I agree, moderators should be immune so they can edit those few questions where "Problem" does belong in the title, but it would still be a good idea for at least a warning pop-up to remind them to edit it out. The filter has forced me to improve lots of titles that I would have forgotten to check when making other edits, but it would be nice for moderators to be able to dismiss it if the title already makes sense.
Moderators should be immune to the word filter. If they can be trusted to handle flag reports, it's likely they already have a history of good quality questions and answers.
This request reminds me of the Futurama line "And I say your three-cent titanium tax doesn't go too far enough!" I support restricting "problem title edit powers," but not to a group as small as diamond moderators.
I think there simply aren't enough mods for this to work. For one thing, mods can't be everywhere. With the number of mods, the number of posts where "problem" titles would be appropriate and the overall number of questions, I have to think the success rate wouldn't be that great. Granted, mod flagging, would help a lot with this.
The bigger issue is that users wishing to edit a question after a special mod edit would be forced to either sit on their hands or re-introduce the error. The former doesn't make the Internet a better place; the latter is inefficient at best, delaying quality and wasting mods' time.
Instead of this, we should give the power to, say, 2k or 5k users. (I do like Bill's idea of a "are you sure" popup either way, though.)