SardineQueen, I agree with your concerns regarding the "inadequate" language that the Deputy Prime Minister has used on this issue. Sex discrimination is definitely much more than his opinion that it is "a little old-fashioned".
A female (or male) cleaner of a royal palace is never going to have as much privilege as a princess. Any change to equalise the rights of females to males in the royal succession is obviously not going to correct that situation but surely it is not even claiming to address that wider issue. It's only a start but we have to start somewhere. Until the monarchy is abolished, someone is still going to end up as monarch, whether the rest of the population are rich or poor, happy or unhappy etc.
Royal women are among the most privileged people in the world but they are still people who deserve equal treatment on the basis of sex, in whatever situation they find themselves.
I think removal of the current sex discrimination in the royal succession would be a positive step. Women's rights should be absolute not relative. I don't want my rights as an ordinary British woman to be reduced or not improved just because there are men and women in other countries who have fewer rights or less money or less privilege.
Removing the current sex bias would send a signal, however limited, that no institution, however high or remote or "traditional", should be allowed to discriminate against a female on the basis of her sex. That is the key feminist point in this issue.
Whoever is sitting on the throne probably does not really have a huge impact on the everyday lives of ordinary people, but I think it is important to ensure that the unfairness of royal succession due to the sex of the royal children is removed. Obviously, that still leaves age discrimination but that's another can of worms that affects both sexes.