Grimarse, I am rather sure there is an exception for relatives. Because I cannot imagine the men actually care for their own elderly fathers. I know for a fact that veils are not required with male relatives, so there's that.
Besides, strict views on female chastity didn't keep male medicine students from killing women with childbed fever by grabbing at their ladyparts with hands unwashed after dissecting corpses. When there's conflicting interests, women will always be required to give up privacy and propriety in order to serve male interests.
It is very rare that women manage to use cultural notions of propriety to gain some advantage. Even safety from sexual assault is something only women who are valuable property to a male get.
If segregation of the sexes really was a top priority, then there'd be lots of highly educated female doctors, lawyers, etc., all the professions where the customer has to be alone with you, and probably also female plumbers (single women are in need of such services, too!), female taxi drivers, et cetera.
That women aren't allowed to drive in Saudi Arabia ensures that if they ever sit in a car, it is with a man.