'Brotherhood of men' would be strange, because it would only refer to adult males. 'Man' is correct, because it refers to the human race. This is a singular noun, and includes all humans of all ages, both male and female. If you use the singular form 'man', without an article, it has the meaning of 'humanity'. For example 'Man is born free, but is everywhere in chains'.

In this case, 'man' refers to the human race, of which there is only one. 'A brotherhood of men' is correct but simply means a closely knit group of males and is in some ways tautological (superfluous), as 'brotherhood' already implies that meaning on its own.

You'll have to ask the original lyricist.

"Man" means all humankind, of both sexes. "The brotherhood of men" would be understood to include both sexes as well. The Barbershop Harmony Society, which organizes and promotes a certain kind of all-male vocal harmony, is a fraternal organization and could be called a brotherhood of men. (Only men can be members; two other organizations, the Sweet Adelines and Harmony Inc. promote a similar style of female quartetting). The Freemasons are a brotherhood of men. Only men are admitted to the society. (There are parallel organizations for women, such as Eastern Star). (Traditionally, in English, groups of mixed gender are referred to by using masculine terms. In recent years, feminists have raised objections and gender-neutral substitutes are being used more and more frequently).