Could you make a run-and-gun shooter with a female character instead of a male?

Why "instead"? Isn't "why would it be female?" on equal grounds with "why would it be male?"? Don't all characters start as genderless, and then you decide one way or the other? Because it's a shooter, it needs to be male unless there is a good reason? (Feel free to tell me if I misinterpreted your words).
I don't want to see female characters only in roles that are stereotypically female. And if I play a female who is a soldier, I don't want a backstory saying that she became a soldier to prove she can be as strong as a man, or something. I want it to be a matter of fact, that of course females are soldiers too, why wouldn't they be?, not even worth a mention.

If being physically fit and strong is a "male" quality, what would be a "female" one? Being gentle and caring, maybe? Does that mean that every game in which you're a nurse/doctor/teacher/creature breeder should have a female lead by default, and that we can ask "why did you make that character male instead of female" when they are men?
I'm sure you can see that there is nothing wrong with gentle and caring male characters, and they don't need a reason to be that way. And they're not "pretending to be a woman", either.

We can have tough females and gentle males. I see how it's a problem if females in video games are ALWAYS shown as needing to have stereotypically male qualities to be considered worthy. Indeed I don't think women should need to emulate males to be as good as them. On the other hand, women with these qualities aren't necessarily emulating men. They might just be themselves, and that's fine too.

There doesn't need to be a divide. "Men do this, women do that." Ultimately, before being a man or a woman, we're people, and we're our own person, with our own characteristics. No man has 100% of the "male" characteristics and 0% of the "female" ones, and no woman is the other way around, either.
I want believable, three dimensional characters. I want to see them express feelings, I want them to be themselves. I want them to take off their armour, wipe off the blood of their friend, and cry. Whether they're a man or a woman.

Actually, it even annoys me when in a game males are melee and females are magic users, for instance. I want the option to be female melee or male caster, too!