This really can't be understated. The difference between your how your brain works when you're under the clock and know you're being observed vs. the "natural" flow of a calm working environment is tremendous. Anyone of us who's conducting the interview must do some kind of double-checking of obvious mistakes to fairly account for the limitations of the process.

Yep! I have been in similar situations where you make a mistake and you get out right rejected. Surely it's better to explain the mistake to me and then we can discuss it. This can also be a nice cultural check too, how well do I respond to my own mistakes? It's a shame most people don't follow your idea here.