I'll say it again. They want the right to be recognised in law under their gender identity in a more straighforward way than with the GRA, with all that entails.
What do you mean by 'widely accepted'? They use the definition of gender identity from the Yogyakarta Principles. These were written by a bunch of self-selected lawyers and academics, and have no legal standing. They may be widely accepted amongst transactivists, but not in the rest of the world.
By "widely accepted" I mean accepted by the EHRC, many democratic governments (66 states in 2008), and the UN is heading that way, too. Basically, and I am sure this is no coincidence, states in which homosexuality and same sex marriage is legal accept the principles, states which deny gay rights oppose them. Russia, the Arab League and Trinidad and Tobago are not big fans, nor are anti-choice Catholic groups.
That's aside from the almost universal acceptance in the medical professionals in western democracies that gender identity is real.
You can say it's not real, but your suggestion it's a fringe view is not plausible. I suppose you could say all these medical professionals are also delusional, or you could accept that there is a fight going on over the definition of woman. It's interesting to see this in action - the wiki definition is "adult human female" plus transwomen and certain intersex people bolted on.
Here is another gender identity defintion from the APA.
Gender Identity: A person’s deeply felt, inherent sense of being a boy, a man, or male; a girl, a woman, or female; or an alternative gender (e.g., genderqueer, gender nonconforming, gender neutral) that may or may not correspond to a person’s sex assigned at birth or to a person’s primary or secondary sex characteristics. Since gender identity is internal, a person’s gender identity is not necessarily visible to others. “Affirmed gender identity” refers to a person’s gender identity after coming out as TGNC or undergoing a social and/or medical transition process