That is often a problem with proposals/design docs in general. In the real job, if proposal is actually required, it would be sent it back with "please add details on UX and how you are going to store email headers". In this case, the proposal was explicitly _not_ required though, so interviewers did not want to ask for more details on the optional document. They checked what was written there, found no problems, and approved it.
No wonder he focused on backend part.
So from the interviewer's standpoint, author was asking about few details of implementation, like "can I use third-party service for email storage?"; and the response was of course "yes, you can" (because assignment was pretty clear that backend does not need to be advanced or even present, and that it's UI that matters)
I guess the question worked as intended, and filtered out candidate who cannot even read the simple requirements.
(The amount of effort was disproportional though, but I am not sure how to solve this in take-home context without discriminating against people who have busy schedules and/or work slowly)