I do not really want to get into this debate in isolation as it is something that I feel is very difficult to navigate.
What I will say, however is that the current Transgender debate has brought us, imho, to the edges of an sociological epoch.
I have been reading up on this debate quite a bit as I am someone who is interested in sociology, history and politics. As a result, what I see with this debate is not as simplistic as to whether a transgender women should compete with biological females (sorry, if that is wrong, I do not intend to offend anyone) but rather the conclusion of feminism/the birth of it's successor.
I say this as feminism was born in the late 18th century/early 19th century with the yearning to bring equality to the sexes. Initially this started with education and then evolved/progressed/branched off into separate strands to other areas such as; the right to vote, hold public office, equal pay, contraception, rights within marriage etc.
Throughout this evolution, their have always been feminists who have agreed/disagreed with each other and others' who supported one strand, then disavowed the other. However, what underpinned all of this was that there is no real reason for there to be differentiation between the sexes as a women could do what a man could do.
However, with the transgender strand, it, imho, brings into question the founding principle of feminism; that equality between the sexes can ever truly be achieved.
I say this, as if you play the founding notion of feminism (true equality between the sexes) to it's end point/it's last strand, it is always bound to end with whether their is any need for for categorisation/differentiation to those born with different genitalia/hormonal differences.
Therefore, I am of the belief that this current debate needs to be looked at as a whole when people debate it. By that, I mean, do you believe that true equality between the sexes can/should exist? If so, then I do not see how one can argue, imho, that transgender people cannot compete in biological female sports or that there is a need for categorisation in sport/pretty much anything. However, if you do support this, then is there a need for a new socio-categorisation/movement?
Equally, if you do believe there is a difference/there is a need for categorisation, then where do you stop? (remember that throughout the history of feminism, there have always been people who agree with one strand and disavow the other.) In addition, is there a need to re-evaluate the founding principal of feminism and acknowledge that true equality cannot ever be achieved due to biological differences or if you disagree with this, then where will feminism go from here with it's founding principal being undermined?
I recognise this is a highly philosophical interpretation of this debate, however I really think it needs to be considered when debating this issue at hand as for me it is interconnected and whatever way it goes, it will ultimately see if true equality between the sexes can be achieved or whether will be concluded it cannot.