I agree with you that metadata for classical music is complicated. Whereas for pop music, it is mostly enough to set Album Title, Track Title, Album Artist, and maybe Track Artist, for classical music, we care also about Composer and Work and many of the Performers.
I disagree that MusicBrainz and Picard are the right players to change in order to make this work better. I think that MusicBrainz has a pretty good database structure for classical music works and composers (and more), while Picard does a pretty good job of placing the MusicBrainz data structure into music file tags. (Note: you should certainly use the Classical Extras plugin together with Picard.)
Instead, you should look for a music player which understands the concept of composer and work and performer, and presents them to you in a sensible way. Apparently Plex does not do a very good job of that.
I play my classical music with QuodLibet. It does an adequate job, and I might eventually write a QuodLibet plugin to make it do a better job.
Some people like Wax.
In any case, I think the problem you identify is real, but the solution lies with the playback apps and their UIs.