I think this is an interesting point, but responding to the last part ("...a lot of the people preaching this gospel of "you shouldn't feel bad when you find out you're wrong" are faking it too. But that's okay.")

I think there are different levels of "feeling bad". When I'm wrong about something, it certainly doesn't feel good, and in some cases it really sucks, but I don't then generalize that feeling to represent something intrinsic to myself. I think the crux is separating the bad feeling of the instance from an overall bad feeling. Ie - I got this one wrong, but that doesn't mean I'm stupid.