I think "Shall we play?" is a closer translation to how this sentence might actually be used.
Nah. "Shall" and "let's," those two words make the sentence imperative, which means the sentence becomes a command. This sentence is just someone asking whether or not s/he and his/er friends are in the state of playing, but in on itself makes no sense xD
Let's is a form of the subjunctive https://web.cn.edu/kwheeler/grammar_subjunctive.html
There may be a false cognate here. In German I would certainly say: "Spielen wir?" and mean "Shall we play?" The question is whether such an implicit request/command works the same way in Dutch.
If "Spelen wij?" is "Are we playing?", then what is "Do we want to play?"