When being trans was fringe maybe this stance would be ok, just like 25 years ago when most gay people were still in the closet. But in today's inclusive society why would a trans person NOT want that to be part of their story? It sure seems, what with the LGBTQ parades and protests and what not, that the trans/gay community is all out at this point.
But beyond that, I think your point on why they want to compete with the cis gender of their choice is incorrect as well. The reasoning we've been told is that trans women want to do everything a woman can do, not that they don't want anyone knowing they're trans. Lia Thomas certainly didn't hide the fact that she was trans, and every time she competed against another trans woman they partied like it's 1999 to let everyone else know they were trans.