As someone who recruits I'd say that I don't think the point is that there are parallels between interviews and other situations and the candidate is being rejected because of that, it's that I just didn't get enough information to make a decision.

Any part way competent interviewer knows the difference between a bad interview and nerves getting the better someone. When it's nerves we're not thinking "they're awful", we're thinking "i don't know how good they are".

At that point I'm generally too busy to try and come up with a bunch of other things myself on the off chance the person will respond better to a different situation BUT if the person were proactive and came back and said "I know I screwed that up, would it be possible to try X instead", I think I'd be open to that.

(That said obviously giving small paid projects work to all potential interviewees doesn't scale particularly well.)