Zebed Habat (aka simchat bat) has been a part of the Sefardi mesora for quite some time. Get a copy of a siddur nusach “edut hamizrach”.
You know, you are arguing to a strawman. The main problem is not that it doesn’t have a sufficient basis. The main problem is the reason it is being done.
As an extreme example, if you find an old minhag to put a fir tree in your living room on Channuka, and you decide to adopt that minhag because you want to fit in with your goyish neighbors, you are still doing something wrong- even if there is some basis. Because of your motivation.
The motivation here is to not accept the roles that chazal and the torah define for men and women.