> [Key signatures] completely obscure the relationship between the pitches, though.
Nope, they exactly allow the notes to show scale degree relationships. Without key signatures, music in any major scale other than C would be very confusing as the regular scale notes would be marked with accidentals.
> I don’t think I entirely understand this, because it still seems so convoluted to me. You have to mentally translate that C to a C, and then translate the C to however that particular note is actually played on your instrument.What does this accomplish?
Same. Because at some point a player stops playing by translating the exact pitch to a fingering (for example), they start playing on a scale. They practice the scale by itself, and other pointless "music" in that key and it's the fingerings of the scale that they learn. I.o.w. this system allows them to think at one higher level of abstraction.
> I’m a bit out in the weeds from here on. C major is identical to A minor, and I don’t understand why we need both.
This whole section reads "I don't understand this so the world must be stupid to have come up with it." Rest assured that C-major and A-minor are not the same. Yes they have the same pitches, but a different starting point (root). A note's relationship to the root of they key of the piece is much more important than the exact pitch. Fur Elise in C-major would sound just like it sounds in A-major. Not A-minor.
> I only looked into this because I want to compose some music, and I feel completely blocked when I just don’t understand a subject at all.
Here's the bright point: You don't need to understand notation to compose music. Just understand scales and chords (and their progressions), and use whatever notation makes sense for you.