I think the problem runs a bit deeper than people thinking they're experts when they're not, though. I think when most people try to diagnose themselves, they look online for symptoms and such. If they already have some conviction in their diagnosis, they might think that they should also have other symptoms when they don't, or start noticing borderline things that they otherwise would easily ignore. Then, when they finally go to the doctor, their list of self-reported symptoms is no longer a list of what they independently noticed but a laundry list of whatever they read on webmd.
I'd be interested in hearing if you think this is actually the case, since you definitely have more experience and knowledge in it than I do. I'm basing most of this off of the personal experience of a few of my friends and the modicum of knowledge I have from a few clinical psychology classes I took in college.