Is gender identity just an internal feeling, a floating node with no effect on the outside world? If so I'm pretty sure there wouldn't be vocal trans activists.

No... What trans people are describing (at least on tumblr) seems to also involve social dysphoria or role dysphoria. The need to associate with the right choice of socializERS, so to speak, which may or may not be the two largest genders of masculinity (= the gender most often associated with natal men) and femininity (= the gender most often associated with natal women). I think gender crit feminists aren't addressing enough the fact or claim that social and role dysphoria often co-occurs with body dysmorphia.

About the socialization thing. Sorry if this comes across as, "What about teh internalizationz?!?!", but I'm afraid that factor is too little acknowledged. Let's provisionally assume that:

if people with male genitals have internalized female socialization (= socialization aimed at females) rather than male socialization, more than what you would "normally" expect given people trying to socialize them as male, then they can claim to be women inside.

Indeed not all internalized socialization agrees with one's identified gender. But I accept trans individuals' accounts about them internalizing the negative gender roles of their identified gender (e.g. trans men internalizing some toxic masculinity), the problem must be with the content of the socialization, not the existence of gender identity so much (the position of mainstream SJ feminism). Also, you may still be able to say that trans men are women and trans women are men due to their genitals but then the gender crit position that trans women cannot internalize ANY misogyny would be questionable.

I see myself as relaying just previously unseen information to you. I hope you address them.