As I see it, there are four questions here, and I certainly cannot answer them all.
- Which is the correct hyphenated adjectival phrase: winner-take-all society or winner-takes-all society?
- If winner-take-all is correct, why doesn't the verb have to agree with the subject?
- What is the verb form called in this construction?
- What is the origin of this phrase?
To my ear, both winner-take-all society and winner-takes-all society sound fine.
Looking at Google Ngrams, we find: .
So winner-take-all has historically been ahead, but winner-takes-all is rapidly catching up. Note that we cannot compare the phrases winner take all with winner takes all using Google Ngrams, because then we would get constructions like ... and the winner takes all!, which are not adjectival phrases. However, I searched in Google books for the phrase with the indefinite article, a winner take/takes all, and virtually all the hits are adjectival phrases.
The only similar construction in adjectival phrases I can think of is dog-eat-dog. Here the construction is definitely dog-eat-dog (confirmed by Google Ngrams, although dog-eats-dog is occasionally used), and the verb also does not agree with the subject.
So I conclude that in similar adjectival phrases (can anybody think of any others?) the verb does not have to agree with the subject, but I am baffled as to what mood/tense the verb is. In dog-eat-dog world it is clearly not the imperative (it's not a command). The infinitive and subjunctive don't seem likely, either.
UPDATE: There's also buyer-beware as in buyer-beware laws. Here, the verb is definitely in the subjunctive, while Henry in his answer has made a pretty good case for the verb in dog-eat-dog being the infinitive. The adjectival phrase winner-take-all may have originally been applied to the American electoral system. However, Google books comes up with a number of apparently earlier instances of bets where everybody contributes some amount, and "winner take all." If this was the origin, it is also the subjunctive, as it would presumably have been shortened from a phrase such as Let the winner take all.