Well put. I would be in the category of "having pain that does not convey information". I've dealt with chronic back and neck pain for a while, and it doesn't seem to have any physical correlate that shows up on MRIs or and standard medical tests. And a whole host of treatments fail to help it, except some high-grade, red-flag drugs. So, the pain doesn't convey any information.

On top of that, the pain makes it very hard to think, which is a pretty obvious extrinsic reason to say it's bad. (I don't know how this would fail to occur to anyone unless they haven't been through significant pain.) Even if the pain is signaling information (about an imminent threat to your health), then it's not being useful unless it directs your attention to the ways to remove the pain.