> Certainly the idea that a score somehow has more information in it than a midi recording is ridiculous

A midi file is designed to contain different kinds of information than a score, so it can absolutely contain less information (of a certain kind) than a score.

Imagine a group of notes that are attacked very close together in time in a midi file - is it meant to be a chord played without arpeggiation? A chord arpeggiated as fast as possible? A chord arpeggiated to a specific tempo?

When a composer writes (say) a chord with an arpeggiation sign next to it, it specifies that the notes are to be played in a certain order, and quickly. There are multiple possible performances that comply with this "specification". MIDI cannot encode this information (the information being the set of acceptable performances).