There's an argument to be made that the Saintess is the legitimacy behind the Crown, given the (I assume First) Saintess was the Queen of Durkis after it was fully established and empowered.
In that sense, it's not so much "the Crown's only legitimized by marrying the Saintess into the royal family", but that the act itself is intrinsic to the Crown. We don't know if there's been more than two saintesses, but if Durkis got its start under one, then the reemergence of another slots right into that notion of tradition that the King and his advisor were discussing at the start of the chapter.
That said - I believe Duke is doing the correct thing, (ostensibly) seeking to avoid a marriage to Liz. Not simply because he should be getting married to Alicia whenever she completes her Spy Games arc, but because he has a vision for the future of Durkis that would seek to shore up and potentially eliminate entirely glaring systemic issues that go beyond "matters of the Saintess".
The fact that power is concentrated in the nobility by virtue of them having potent magical capability is in itself a problem--as Gil points out, magic isn't an inherent source of geopolitical might, and Durkis' saving grace all this time has been the relative ignorance of its neighboring countries. As things progress in time, it'll become harder and harder to keep that secret from getting out, and Durkis is apparently woefully unprepared to weather any concerted showing of force by any rival nation.
The notion of Liz and a marriage into the Crown is important, precisely because of the precedent set by way of tradition within that nation. But I think it stems less from a question of legitimacy where the Saintess is concerned, and more Duke's vision that Durkis' continued adherence to traditional practices simply because of their established precedent is fundamentally insufficient as time continues forward.