Feminists who believe that feminism is about liberating every person from every type of oppression can unconditionally embrace all trans people, all trans rights and belief in the concept of gender identity ("I feel like a woman therefore I am a woman"). I'd be very surprised if these feminists did not do this.
Feminists who believe that feminism is about liberating women and girls from the system of oppression called gender, which puts women and girls at the bottom of a gender status hierarchy because of their female biology, usually cannot unconditionally embrace all transwomen and all trans rights. I think these feminists can, and do, support improvement of many trans rights e.g. employment rights or freedom to be safe from violence. Their support for trans people runs into difficulties when trans rights and women's rights conflict. I call this gender-identity-critical (GIC) feminism.
The political ideology of transgenderism is about the colonisation of women's space by a subset of male people. This is incompatible with GIC feminism which centres the rights of female people.
Transgenderism is not like GIC feminism or some other well-known social movements. e.g. civil rights and gay rights, in which the activists campaign to be acknowledged as human and equal to the oppressor. Much of transgenderism is about appropriating the space of another oppressed group (women and girls) and removing or at least trumping their rights.
Extremist transactivists campaign for transgenderism, not the human rights of trans people.
Has the GIC feminism versus transgenderism debate already passed the point of no-return? Currently, GIC feminists say transwomen are never women, transactivists say transwomen are always women. What should a debate about women hope to achieve? Perhaps a compromise that agrees that all transwomen are women is some ways. e.g. no transwomen in women's refuges or women's sport? Perhaps a compromise that only some transwomen are women e.g. only castrated transwomen who take hormones and who look "pretty" enough?
A shift in emphasis by GIC feminists would be very difficult because, as many MNers already know, GIC feminism is already seen as bigoted and transphobic. I think this is a highlight in the achievements of patriarchal backlash. It's the perfect storm for women's rights. Acceptance by society that transwomen are women is the perfect patriarchal tool. It is enabling the removal or restriction of rights that women and girls have fought for on our path to liberation from oppression that is based on female biology.
At the moment, I think a major problem for GIC feminism is the transphobia and bigotry labelling of anything short of total unconditional agreement with the transactivist belief that anyone who claims to feel like a woman is a woman in every way. Therefore, I think the only way for GIC feminists to be seen as not transphobic and not bigoted is either unconditional surrender to this demand or a drastic reduction in the scope of what can be labelled as transphobic or bigoted. Total surrender would make debate unnecessary so I'd prefer the starting point to be a redefinition of transphobia. Only then do I think any constructive debate could begin.
A huge obstacle to overcome is the current deadlock over who is a woman. GIC feminists insist that transwomen are not women, because they are not female. I don't see how framing the debate around women would free it from the current deadlock, because if a person thinks that transwomen are women, then all debate around women and gender will still centre transwomen because, according to transactivists, transwomen are the most oppressed group and lack the "cis-privilege" of "other" women. Even if the debate is changed to be about female people instead of women, transactivists just play their "transwomen are female" card.