The bracha "Shelo asani isha" is often defended by Orthodox apologists as being about mitzvos. It's not about men thanking Hashem for not making them someone of lower status, but about thanking Him for not making them someone who had fewer mitzvos. Non-Jew, slaves, and women have fewer mitzvos than men, and so men thank Hashem for not being in a category of people with fewer mitzvos to perform.
I don't think this explanation works.
1. If it is something to thank God for not being, then that means it is undesirable. At the least, the bracha implies that being a woman is less desirable than being a man.
2. It is part of a group of brachos thanking Hashem for not being afflicted. We bless him for opening the eyes of the blind, releasing the bound, straightening the bowed, clothing the naked, etc., giving the Jewish people might and glory, and not making us non-Jews, slaves, or women.
3. If it's about mitzvos, why do women say the bracha thanking God for not making them slaves? Women and slaves are obligated in the same number of mitzvos.*
4. In the same vein, why is there no bracha for kohanim, "Shelo asani Yisrael?"
*Does anyone happen to know offhand where the gemara says this? I found a reference to gitten, but it was four blatt, and I'm not in the mood to comb through it.