I've always reckoned phrases like 'a beautiful king's daughter' as [a beautiful king's] daughter, i.e. it is the king who is beautiful, but have never considered the phrase can give the information that daughter is beautiful.Moreover, in the s-genitive in modifier relation any further premodification can only modify the possessum, but not the possessor, i.e. it is the daughter that is beautiful (20a) and the license that is old (21a). In contrast, in the determiner reading it is the possessor that is modified by the adjective, i.e. it is the king who is beautiful (20b) and the driver who is old (21b).
a. a beautiful [king's daughter]
b. [a beautiful king's] daughter
a. the old [driver's license]
b. [the old driver's] license
Source: Genitive Variation in English: Conceptual Factors in Synchronic and Diachronic Studies by Annette Rosenbach
Do you really perceive 'a beautiful king's daughter' as either 'it is the king who is beautiful' or 'it is the daughter who is beautiful'?