First, I want to be clear: my thoughts aren't "this is exactly the way video games need to be or I'll be mad", they're "I want us to rethink why video games are the way they are". So, I'm not saying my opinions have to be law, or that they'll always be right.
But my first question would be, why does it need to be a "nondescript male" in that role? What does that character trait bring to the character or the game that could not be done if the character were female instead? Is it because it's the story of a father who has lost his child, or a husband who has lost his wife? Okay, if so, are you telling that story because you've got an actual story to tell, or because that's just the easiest cliche to go with?
On one side, let's taking something like Resistance 3. The hero was a nondescript male, but there was never one storyline beat or moment of importance (that I can remember) that really built upon that idea. If I were to magically make him a her, the story wouldn't change, and the character's progression through the game wouldn't be affected any. I could have had that choice, and the game wouldn't have been the lesser for it.
On the other side, you've got games like Silent Hill 2 and The Last of Us. Both build upon actual, crafted stories, where who those characters are is absolutely important. James was a husband trying to find his dead wife, and him being the husband and her being the wife were key elements to what was being told. Or Joel, where what he's been through, and the relationship that he forms with Ellie, probably wouldn't be as strong if you just let the player decide who Joel is supposed to be.
I'd argue that your statement works against the idea that the main character of the game has to be male in COD. Why can't a woman perform heroic deeds? Why can't she be the action hero? Why can't she go up against an entire army? Nearly everything that happens in COD's single player is completely unrealistic, so putting a female into that position changes nothing. And if my character is "nondescript male", then that pisses me off even more, because that means that there really is no reason that I couldn't be given the option to make generic white dude into anyone I want them to be.
Because it's a mental thing for me. In single-player games, I enjoy the fantasy, and almost never do I try to put myself into the game. In multiplayer, it's different. Multiplayer is me, competing with others. When you say "you're going to make a soldier, and customize them, and configure them, and that soldier is going to represent you on the battlefield", and then I see that solider, and it's someone who doesn't represent me because of its gender, while the majority of other players have soldiers that do represent them, I feel left outno matter if that makes sense to someone else or not.
And I have far too little free time and far too many gaming choices to play something that makes me feel left out.
That's not to say that I will never play a multiplayer game that lacks such choices, because I've absolutely had examples of putting up with a game's lack of choice due to my enjoyment for the game (The Last of Us' Factions mode being a great recent example). At this point, though, there are enough games out there that understand the importance of choice that I can simply avoid those that don't.
And let me ask this: if gender choice in COD's multiplayer truly doesn't matter, then wouldn't it be true that the next game could come out, force all players to only play as female soldiers, and that should be okay? What if the tables were turned?