And as a complete non-expert in this subfield of math (or anything within a hundred miles of it) I would love to hear why people are saying stuff like "equality is a fundamental mistake of mathematics and this approach is fixing it" rather than "this approach is developing a more refined/general/flexible notion of equality".

After a few minutes of deferential throat clearing, I'd like to say "in all seriousness, come on, equality isn't a mistake, and whatever brilliant insight your theory affords still isn't going to do away with it."