The Israeli charedi parties represent the right-wing of charedim. The “new charedim,” more moderate charedim, the “left-wing yeshivish,” American-olim who are charedim, etc., probably have a somewhat more open attitude toward women’s occupations and public positions.
There are quite a few charedi women who are doctors, lawyers, professors, etc. in America, and at least some in Israel. There are certainly charedi women in both countries who write books and articles and give public shiurim (to women); some even produce and direct their own films (for women only).
Something else to keep in mind that, according to some Torah teachings, everything a man accomplishes he owes to his wife’s merits. So perhaps there’s no specific need, in people’s minds, of women MKs, since women MK’s wives are ultimately responsible for all the good the MKs are able to do (either directly, through conversation, or in some mysterious spiritual way).
However, I agree with you that there is no real fundamental reason why women, including charedi women, couldn’t run for office. I think they will eventually, but perhaps not as part of one of the current charedi parties (because they wouldn’t want to do anything to offend the community’s kannoim), but instead through Likud or as an independent or something.
Some people would see the tsnius issue as being in the limelight where everyone can see you, as opposed to behind the scenes. But not everyone would agree with that interpretation of tsnius. Nowadays charedi publications don’t publish pictures of women, but that’s no reason not to elect them; they can just run stories in their newspapers about the women politicians without pictures.
Do any American Orthodox women hold political office? Just wondering.