I have an anecdote of a time I received feedback for a (non-EA) job I didn't get in a manner I found helpful. Hopefully this helps hiring managers who are brainstorming ways to provide feedback.
I had applied for a consulting position as an undergraduate. I was selected for a second-round interview, where I was asked to discuss how I would advise a client (IIRC, the client was a grocery store considering a merger). A few days later, my interviewer called me and said, "Sorry, you didn't get the job. Would you like some feedback on your interview?" I said yes, and he proceeded, "We were looking for a response that addressed factor ___ and you didn't mention that."
What I liked about this was that (A) I received feedback, which is rare in hiring processes and (B) the feedback was specifically about my interview answer, not about broad qualities I was missing. The latter made the feedback easier to manage emotionally, because I didn't end up thinking "oh no, I'm not creative/thorough/skilled." Instead, my takeaway was "I should talk about more facets of the problem if I do this again."
I'm so happy that people enjoyed reading this. This review is written in a lighter tone than a lot of other posts, so I was half-afraid I was committing a social faux pas. I'm glad that's not the case.