It doesn’t apply to babysitters (in your scenario) because it is your decision, and the assumption is that your decision is solely based on who can best do the job.
Now, certainly that should be true for public office as well, but it clearly isn’t always.
Do you think it’s coincidence that the US hasn’t had a female president, and has had only one black president? I think we made a decision on the current president based on race rather than qualification, and probably in the past, there could have been better presidents than we had, had race and gender not been a factor.
So, while I agree that you can’t argue about a particular person that she has a right to be selected, it might be fair for a society to make judgments based on the broader picture that a class has been discriminated against.
And you didn’t answer my question, but that really gets to the heart of it. The unspoken accusation here is that there are no female MKs in the chareidi parties because chareidim don’t respect women.
Your argument, that nobody has a right to a seat, doesn’t address that. Mine, that there are legitimate reasons unrelated to respect, does.