I'm not sure it is an error so I'm asking the question here instead. Shouldn't this be "Αυτός θέλει να παίξει."?
"He wants to be playing" should be accepted as correct. If not the only correct translation.
If we use infinitive "to play" it depends from the pronoun ? for example if it would be "we want to play" then it would be εμεις θελουμε να παιζουμε? Thank you
Modern Greek doesn't have an infinitive -- at least, not one that works the way the English/French/German/etc. one does.
So you have to use a finite verb in this sort of situation -- the literal translation would be something like "He wants that he plays".
And yes, "We want to play" = "We want that we play" = (Εμείς) θέλουμε να παίζουμε.
There's also no gerund, so something like "Smoking is bad for you" or "Playing tennis is fun" or "Using drugs is dangerous" has to be recast into something like "the-fact-that-you-smoke is bad for you" and "the-fact-that-I-play-tennis is fun" and "the-fact-that-someone-uses-drugs is dangerous": you have to say who's doing the action because you're using a finite verb.