You've brushed up against a key point, I believe. Given a set container all of the problems an interviewer could ask me to solve, there exists a subset of those problems that I've solved previously, but that subset is small.
The problem here is that I don't believe most engineers can effectively explain their thinking when they are still working on solving the problem. Perhaps this is something I could work on, but I prefer everything to be silent while I gather my thoughts. It is my understanding that being quiet when asked by an interviewer to solve a given problem is a bad thing, which ultimately leads to points being docked because I have a process for considering problems that works very well for me.
Conversely, should the interviewee be free to pick a problem they've previously solved and talk with their interviewer about it, they should already have their thoughts gathered. Working under the assumption that the candidate put in a substantial amount of work, they should be able to talk at length about the project. An interesting and useful consequence of this is that, in the event a candidate can't speak about a project they worked on, I believe that says something about their capabilities as an engineer. Should the interviewer be put off by this? Maybe. Maybe not. How far down the rabbit hole do you want to go?
Overall, I think this is a pretty interesting problem that could use some good discussion to generate ideas about what a more effective and efficient process might look like.