I think it works that way in German, French and Spanish.
Nope, German has the differentiation between his (sein/e) and hers (ihr/e) and agreement with the gender of the object. Gendered pronouns are about specificity: whose ball is that?. If the pronoun doesn't provide the information then something else in the context will have to. Or the audience will ask. My Swedish teacher says that in Finnish (she's a Finnish Swede), which is famously ungendered, it is common to ask whether a man or a woman is the subject, because it is often not clear from the context.
Back to romance languages: the absence of gender of the possessive third-person pronoun can be set against the use of gender in the third party plural personal pronoun, at least in French: ils they (male), elles they (female). Then again, personal pronouns are less important in romance languages than they are in Germanic ones, French being an odd mixture of the two.