I find this tendency a lot when writing, that a character I hadn't originally intended to make gay or who I hadn't planned on thinking about their sxcuality at all suddenly just revealed themselves to be naturally gay. It wasn't like a stereotype that I gave them or even a conscious decision, it's just that when you spend a lot of time using a character (either that someone else has written or that you're writing) it becomes almost instinctual. 'Is Shepard a lesbian? Of course she is, I don't even need to think about it, she just is.' (For FemShep/Liara, which is basically my go to setting)
And if I play the character differently deliberately, then sometimes it changes. When I played my renegade FemShep with a ruthless background it just didn't make sense that she would get with anyone, because all of the options were just not tough enough for her, whereas for my Paragon who automatically tried to make everyone she encountered feel better, Liara needed some cheering up dammit!
I think that a good story well told (I'm not saying my writing is good but I like it so to me it is good story well told) will leave just enough blanks that you fill them in yourself, and this is why the videogame is such a perfect medium, because it can allow for every possible outcome of a blank being filled. Is your character good or bad or neutral? The game designer can fill in the blank, if your character introvert or extrovert? The game designer can give you an experience to fit both.
There are of course technological restrictions and there is still a lot that we can't do, but that the potential is there for people to almost write their own stories as they play is incredible. I don't think there'll ever be a desire for only open games with no linearity, but I think it's always good to be able to take a step back and say 'look how far we've come.' When I started gaming you could make a blocky representation of a woman climb things and shoot tigers, nowadays I can make a nearly photorealistic man have a deep and meaningful relationship with any number of characters I choose. It's like comparing a cave painting to the Mona Lisa, or The Roundhay Garden Scene to Black Swan.