I have mixed feelings about the idea: yes play is important, but is thinking of it as a separate mechanism the right modularity? Animals play, so we know that it is not a cultural invention but something more basic, and since we don't have very good theories about that kind of stuff we invent the idea of a drive (trieb), but it might be so much phlogiston if we really understood how behavior worked.
Or: not sure play is so much a drive as a consequence of the interaction of other drives. It doesn't quite make sense as a drive, it doesn't have obvious survival value as an end-goal. If sexual competition leads to status competition leads to games – well, ok, there is a evpsych just-so story for you, it explains play but also explains it away as a behavioral primitive.
But a theorist of deep play would reply, I think, that that is a crude way to think of play, which can have elements of competition but is not really about that. That it is gratuitous to survival and base goals is part of its point, it is what permits the emergence of higher goals.
Culture, in other words, has its own drives because it is an emergent layer on top of our basic ape selves. And at the level of culture and persons, Spieltrieb is actually pretty basic.