It is kinda at a certain level in your career you are expected to just know how to do it. For other skills like management, you at least have your peers who are doing the work day in day out, who you can learn from. Interviews are closed room, usually, though I have had some where one person was shadowing the interviewer.
Also, if you take a terrible interview, its not gonna really impact you. There is no performance report that'll negatively impact you. There will be another candidate, who you'll have a rapport with. Or someone else will interview someone else.
As you said in your anecdote about the binary search problem, not everyone is best at being an interviewer.
But aside from there being a dedicated person for interviewing for big companies, or outsourcing interviews for startups, I don't really see a solution. And outsourcing, whether internally or externally comes up with its own set of problems.