Language use changes. When those words were written in 1776, the term “men” included women for some purposes at least.
In modern English, it is politically correct to adjust words and grammar to avoid any hint of sexism. There is nothing wrong in doing so as long as it is done properly.
One way is to exchange the man in a word with person or a more general word. For example, a meeting is run by a chairperson while the firefighters and police officers deal with the riot outside. In none of these cases is the structure of the sentence changed or rendered clumsy.
For pronoun reference to general groups of people, using the plural noun avoids the gender issue. For example, students that fail their test will have a second opportunity. That is far more natural than a student that fails his/her test. Sometimes, the context does not allow that option. In that case, my personal preference is to write s/he. In legal language, there is an additional trick: define the individual as a party of some kind. Then, the writer can use the term it and dispense with the various forms of he and she.
Other languages handle the issue differently. Russian has the pronoun свой which refers to the subject, whether that is first, second, or third person. So, at least in the sentence, after using a noun or pronoun as the subject, the writer can avoid the his/her construction. Another advantage of this pronoun is that avoid a classic problem in English with two male or female persons in a sentence and one possessive pronoun: My father and uncle argued about who had taken his wallet. Whose wallet are we talking about, the former’s or the latter’s? French places grammatical gender as the main criterion: Madame Le Juge. Note that Madame is clearly feminine but le is masculine because the noun juge is masculine. Hebrew, coming from the Middle East, simplifies the matter: if there is one male in the group, a male pronoun is used, period. After all, all men are created equal!