The players are not the subjects of play; instead play merely reaches presentation (Darstellung) through the players. We can already see this from the use of the word, especially from its many metaphorical uses.
Here as always the metaphorical usage has methodological priority. If a word is applied to a sphere to which it did not originally belong, the actual "original" meaning emerges quite clearly. Language has performed in advance the abstraction that is, as such, the task of conceptual analysis. Now thinking need only make use of this advance achievement.
Something like this methodological priority seems to me to be what I have been thinking towards in my analysis of having, except that I do not limit it to a merely methodological priority. I particularly draw your attention to that last-linked posting, in which I insisted that having an ability more originally belongs to having than does having in the hand. True: Gadamer does say that the 'actual "original" meaning emerges' only in the light of the later transference of the term, and this might mean that the original usage was secretly guided by something more original. But this saying should only lead us to ask what it means for a word to have an "actual" meaning, when actually no one means that, yet.