Regarding the facts around what is going on rather what ought to be done I'm pretty confident social aspects of gender are not all a social construct.

All societies have social forms of gender. Masculinity and femininity. Men are masculine and attracted to feminine women and women are feminine and attracted to masculine men.

Biology makes men and women differently, that is likely to include behaviours to utilise the differences. But humans being as flexible as they are the exact forms are completed by culture.

There are a constant minority percentage of people across all societies who do not strictly fit that pattern. Which implies a natural trigger not a pure social construct.

Is it mental illness? It depends on your definition of mental illness. So in a sense defying social norms is mental illness. But people in this category are not necessarily disconnected from reality.

I have less evidence for, but guess at a component hybrid model of gender. That some parts are associated with a sex but the combinations do not create perfect averages or perfect mirrors. For example I think same sex attraction IS from an opposite sex behaviour, however the combination does not always result in the same behaviour.