While signs like "Men working" may be factually incorrect, they are easily fixable by replacing the noun (for example, with "people") or by rephrasing the sign.
It's just another of many examples where the noun was replaced by a more inclusive one (firemen -> firefighters, postmen -> mail carriers, sportsmen -> athletes, etc.) It can be done without bending and breaking the grammar rules - and, thus, probably, should be done.
But in case of the pronoun, it is not easily replaceable by a grammatically correct AND not overly heavy version (which is also not an artificial construct) - and, I think, it shouldn't be. Language is language, grammar is grammar, and I also have never seen an issue with using "he", "him", or "his" in a general case which may be applied to a person of either gender. Please, no "he or she", "they" or "ip"!
I noticed, many magazines, say, those for parents, started to alternate "he" and "she" when talking about "a child". Say, in one paragraph it's "If your child has cold, give her plenty of liquids to drink", and in the other "If your child hurt himself on a playground, do such and such".
I came from a language background where all the nouns are either feminine, masculine, or neutral - grammatically, not by meaning. So maybe that's why I wouldn't have a problem if they were more consistent referring to a child (or a person) as "he" everywhere - not to imply that your child necessarily a boy, but just because I am used to it, because for me, word "child" have always been masculine grammatically.