Why do some people hold a woman should not say kadish? She's not singing so it shouldn't be Kol Isha, and if she's behind a mechitza it shouldn't be a tzniut problem.
R' Brody discusses the issue on JPost's Ask the Rabbi. He cites many authorities, such as R.Y.D Soloveitchik and R.Y.Henkin, that permitted it and R.M. Feinstein who said it was common practice in Europe. However, he mentions some objections raised:
The permissive position was opposed by many decisors, who argued that (a) women could not participate in this part of the service (Minhat Yitzchak 4:30), (b) the classic mystical sources only spoke of men’s recitation (Mishpatei Uziel OC 3:13), (c) it was immodest in public settings (Aseh Lecha Rav 5:33), or (d) it would ultimately support antinomian trends found in the non- Orthodox movements (Yahel Yisrael 2:90).
It may be that in certain areas the Reform and Conservative movements cause an opposite reaction from Orthodox Jews.
Rav Moshe talks about this in Igros Moshe OC(5) 12:2. He says women did say kaddish but the tshuvah is more focused on a mechitza for one or two women .