What I would suggest is that when the interviewer and candidate get reach the "quiz or coding exercise" part, have them pick a question from a website that provides a question at random, and let both work together towards a solution.
This matches more closely with what they will end up doing anyways if the candidate is hired, and will remove the "I've seen 20 different ways to solve this" bias while also generating empathy for the candidate when the interviewer him/herself also struggles with a fresh problem.
Of course, the interviewer can try to lead the candidate and can hold back from telling the solution right away if s/he can clearly see it, but if not this could set the stage for a fairly realistic way to sample all kinds of qualities, from technical, to communication, to empathy, to teamwork, etc...
I think it would also be less stressful from the candidate's point of view: a candidate often feels like an interview seems unfair because s/he is being asked about something that the interviewer has had a chance to review and prepare much in advance.
When interviewers ask trick/complicated questions, I sometimes wonder what would happen if they were to ask the same question to the rest of their own team. Are they expected to answer it well? Would they know? Think about the places you've worked and the questions you've given at interviews: do you think your own colleagues would have aced them?
Obviously, this doesn't completely remove the additional stress on the candidate, since the interviewer's job isn't on the line, but I think it would provide a more balanced assessment of the candidate's abilities and personality traits.
In practice, I'm not sure interviewers would be very receptive to such method, as it could turn an interview into a stressful event for them.