Are you planning to create tech management interview cases? They don't seem to be as qualified as the people they hire. Seems like a problem in need of better application screening processes.
Also, the best engineers aren't necessarily the best engineering managers. See https://en.wikipedia.org/wiki/Peter_principle.
We don't believe tech management interviews should at all be automated. You really want these to be as qualitative and human-driven as possible.
It is a convenient myth that people who build the actual systems can be put through an automated filtering process capable of sifting wheat from chafe where as those who manage them are doing subjective, qualitative work that cannot possibly be measured.
I do think everything can be measured given the data, and hence automated. The problem in the latter case seems to me that the data is not easily accessible.