"We don't have weak people; we are all very strong"
Translation:У нас нет слабых, мы все очень сильные
Is there a reason why "people" is omitted in the Russian? Just a style choice?
This is similar to a very, very common structure in Russian. That is, the noun common to two (or more) contrasting adjectives is most often emitted in the subsequent part(s) of a comparison or contrast. In this particular instance, the first noun "people" is implicit in the pronoun "we".
By the way, the "we" in this sentence must be inhabitants of Lake Wobegon, "where all the women are strong, all the men are good looking, and all the children are above average." (https://en.wikipedia.org/wiki/Lake_Wobegon#Standard_monologue_items)