I feel like you are conflating “the different combinations of directly reproduction-related traits which can occur in species which have sexes” and “the kinds of viable gametes which exist in creatures that have sexes”.

It seems like your definition of the sex framework is based on “the kinds of viable gametes of creatures which have sexes” (I do think this is a binary, let’s call this X), while other people are arguing for a definition more like “the combinations of traits in these creatures of which certain combinations are directly responsible for the creation of viable gametes” (there are more than two of these, and it’s not clear how to enumerate them).

That said, I might be wrong about what you are arguing and what other people are arguing. I’m sorry if I’ve wasted your time in some way.

Also (a pedantic complaint) you said these things:

they are all based around a universal binary biological structure

The bio-chemistry of terrestrial life is built upon a binary sex framework

There are multiple species of terrestrial fungi which use “mating systems” which aren’t sex-based and aren’t necessarily binary.

Again, I’m sorry if I’ve wasted your time in some way.

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