No it makes no sense at all. The entire reason for this discussion is that people do not - any longer - feel comfortable using he that way either, so a pronoun that is - to the ear as well as to the eye - non gender specific is being sought. To use she and her would be no improvement over he and him.
There are
some people who do not feel comfortable using
he and
man to express the concept of people in general, since it means using masculine words to identify a person who might be female. There are other people who have no problem with that at all, having been raised to believe that using
he and
man are the grammatically proper way to identify people in general.
Similarly there are
some people who do not feel comfortable using
they,
them, and
themself to express the concept of people in general, since it means using plural words to identify a single person. There are other people who have no problem with that.
In other words, neither of these is a perfect solution.
Another option, also imperfect, is the use of s/he. I did not claim it made perfect sense; I said it made more sense than the use of he.
And it does. He both sounds identical with and looks identical with the pronoun used to identify males; s/he sounds identical with the pronound used to identify females, but looks different. It visually includes both male and female. Since what is desired is a word to identify a single person who might be either female or male, that makes more sense than the use of a word which visually includes only one sex.
S/he is socially not a workable solution, because people in practice aren't able to pronounce it comfortably. Therefore in practice
them is probably a better solution then
s/he. But since there are people who are not comfortable with
them since it's plural, I think it's worth looking at and considering s/he in trying to understand the various possibilities and in trying to understand why different people accept some possibilities and reject others.