Gender is a role, typically assigned because of one's biological sex. It is only as valid as society dictates and can be fluid, as especially women have proven since the 1960s.
Unlike you, I welcome trans women as equal to my own status as a woman and feminist. I personally know two trans persons, who have transitioned to female, and they are sincere in their desire to live as women.
Of course, I do understand the concern about fairness in sports that you mention, and specific guidelines to protect women's sports need to be considered.
I don't understand your animosity towards trans people. They are hardly the ones leading to our objectification. Both men and women cause women to be overly objectified. Women need to use legislation and education to be treated as more than our appearance and other superficial measures of our worth. However, there is no need to bash trans people to accomplish that goal. I believe we can do that together and united.