The gender of the user of the word ought to be unimportant, but it adds to a context that determines how it's being used.
If we generalize what you've said to race, it shouldn't matter whether a white person is using the n-word. Except some number of white people did use that word for a long time with utmost contempt. If you used that word as a white person in an inclusive workplace today with the full contempt of the word, I doubt you'd still be in post 24 hours later unless you had something like tourettes. [0] I doubt we'd be having such a discussion about that.
In many cultures, women have been treated as second-class citizens and, in many cultures, they are still treated as second-class citizens. Consider that in one canton in Switzerland, women didn't get to vote on local matters until 1990. [1]
In nearly all countries, the distribution of power is still very much skewed towards men. The difference between a woman calling another woman a bitch and a man is the potential for confusion with contempt based purely on gender alone. A woman is perceived as unlikely to hold contemptuous views towards women based on their gender alone. This doesn't mean there are not women who despise women for being women, it's just that it is not perceived as such. It even depends on where it is and how the word is used. As such it's probably one of those words to avoid.
This isn't political-correctness-gone-mad or because of some liberal cultural-marxist conspiracy to turn men into second-class citizens; it's because in a fraction of a second, even though that word may have no connotations of contempt to you, you have potentially reduced that person to an object of that contempt. Some people may not be affected by this, but some people will and be rather hurt. It's simply bad for business, no matter how useless you may perceive that person to be. If men had been treated as second-class citizens for centuries by women and dick was a pejorative term used against men, it would be a problem if a woman used that word against a man.
Consider the following:
- You'll never guess what that bitch has done now.
- You bitch!
- She's a bit of a bitch
- Hey, bitches!
Where I'm from (UK) each has a different level of contempt. I can't see that much in the last one. The first one is almost certainly contemptuous. The second one is very context dependent: if said scornfully it's contemptuous; if said jokingly it's probably not, although it could be confused as such.
[0] Although, if you did have Tourette syndrome or similar, it is unlikely that you would say it with the full contempt unless you actually felt that contempt. Sadly, some people with tourettes do find it hard to keep a job because it is rather poorly understood.
[1] https://en.wikipedia.org/wiki/Women%27s_suffrage_in_Switzerl...