In this sentence, does "than his father" modify the entire phrase "better player"? He was a better player than his father.
than his father is not a modifier but the complement of the comparative adjective better.†
The comparative construction -er .. than his father modifies the adjective good, and the resultant AP better than his father modifies player.
† If you want to get really fancy you could say that it's the complement of the comparative suffix -er (just as it would be the complement of the comparative adverb more in more active than his father), or that it's the complement of the 'feature' comparative grade written out as -er.