So much history here, and so many well-informed people. It's unusual to see an informed and yet civil discussion on modern events on any internet forum. The conclusion I draw is that many people like the tradition, pomp and predictability of having one family, with all its familial convulations involved, as at least part of the state.
But most people do not want biological roulette to choose their leaders.
DNA testing could play a role in some people's thinking, of course (and there are many ways to do it, and more of it being done every day). Monarchs have been known to adopt or legitimize offspring that were not biologically their own; sometimes it was a cause for disruption among the people (who usually have no way of knowing), mostly it's only known way after the fact (or speculated about).
The emphasis on male line, and the rejection of adoption as a legitimate way of family building seem to me to be fundamentally against the way most French (and Americans) think these days. Monarchists are described as conservative for many reasons - these are two of them.
A person doesn't have to donate an egg or sperm to a child to consider that child their own; women are capable of ruling.