The legitimacy issue is perhaps more important. As long as some people with influence accept the theory that gives legitimacy to the rightful heir, the monarch has a built-in advantage over any challenger. That may be quite small, but there is a multiplier effect, because those who value stability and peace will also support the monarch over a rival in order to preserve that stabilising factor. That is quite explicit in some writing from the period (nothing to hand, sorry) - "We must support the King because the existence of a rightful king is what saves us from perpetual civil war"

Monarchies tend to break down when that advantage is lost, either due to unclear succession, or an obviously incompetent heir. Civil war due to these situations was common enough that the stability argument was seen as realistic and concrete.