Leaving aside current views on gender identity, historically, "man" has been used as an umbrella term for both genders - and it still is, unless someone objects to it. "Mankind" refers to all human beings (although the term "womankind" has been coined from this to denote only women). When Neil Armstrong said "one giant leap for mankind" he was referring to all humanity, not just males. The term "man-made" refers to anything which has been engineered by a human being, and not by nature.
When referring to specific individuals, however, we always use the correct pronouns, but there has been a historic preference to default to the male gender pronouns when writing certain kinds of documents, such as instructions, that would apply to both genders. For example:
If the user receives an error, he should report it.
Some would avoid this by using the gender-neutral "they"; however, another reason such writing uses a specific pronoun in instructions is to firmly place responsibility with individuals, and using "they" can sound more passive as if the instructions apply to someone else. The alternative is to write:
If the user receives an error, he or she should report it.
The idea behind defaulting to one gender is that it saves time and space, and makes the document more readable. It is quite common for a document or article to state from the beginning that they will use the male gender, but that it applies to both. For example, a popular parenting book refers to the baby as "he" throughout but notes that all advice applies to both baby boys and girls.
In any kind of writing, if you use an individual as an example and suggest that this example applies to a wider range of people, you would use the gender terms appropriate to that individual, and although you are not actually using those terms to address other people of different genders, what you are saying does by extension apply to them.
So the answer to your question is yes - the terms "man", and the associated pronouns can in certain contexts refer to all humankind.