> tl;dr auto tagging is broken, manual tagging is necessary for correctness
Well, ideally, you would enter all relevant information into a global database and simply configure your tagger to store the tags based on your preferences; that way you don't have to do redundant manual work and benefit from data that others have entered.
> I have manually tagged it to indicate which operetta it was from and in what year it was originally produced
> giving it a title based on how the album named it (e.g. "Pirates of Penzance Major General's Song" instead of the correct "I am the very model of a modern major general"), and setting the year to when the CD was released
Of course people have a different understanding of "correctness". The MusicBrainz schema is fairly flexible, there are three levels of detail:
1. Track as labeled on an album track list
2. Recording, which links all different releases/tracks of the same recording
3. Optionally, there's "work", which collects together recordings of a single work, but can also relate to other works such as the opera.
Sounds like what you want is to actually store the attributes (title and date) of the work (#3) in your tags, not the track list (#1). Now, Picard itself doesn't support this, but my point is that MusicBrainz isn't like the CDDB databases that only knew about track lists. MusicBrainz a first-class music database and the community actually cares about correctness. With some additional code it would be possible to fetch all the relevant details you care about.
Not all editors enter information at this detail -- and earlier versions didn't have all these capabilities. But that's a small matter of going in and entering/fixing the data, which you already have to do when building your directory structure anyway.
> or setting the genre to "Comedy" instead of "Light Opera"
MusicBrainz doesn't support genres as such because (as you point out) people often have wildly different understanding of genres. But it does provide a "folksonomy tags" feature. Picard can be configured to only accept genres that you personally have entered, or set a threshold on how popular the tag has to be to be applied.