i do support your view that our society still need gothram and that a divorced woman could claim the gothram of her ex-husband.
at any cost she cannot revert back to her father's gothram because when marriages are solemnized, as per our tradition, the girls father says he hands over his daughter born in his gothram to the bridegroom's family and the bridgroom's father accepts her into his family by declaring that from that moment she belongs to his gothram.
The divorce was not sanctioned in our 'dharma'. So it is new situations and needs a new solution. If the ex-husband wants her not to use his name as her last name, it can be true for Gothram. I do not know the need for a Gothram for her. If it is for re-marriage then that complicates the problem even more.
My contention is that there is absolutely no connection between linage and Gothram. If there was one at some time that has been corrupted and has no value in the present day world.