Your post is raising plenty of thorny issues, so bear with me, and if I make any mistakes that offend anyone reading this, understand they are mistakes and do not stem out of maliciousness.
Anyway, I'll start with saying that the video linked at the start of the post does not impress me. So you say the Time Lords aren't humans and we shouldn't keep ascribing human societal and biological functions to them? Great!
So, why are you then, after saying that this will be futile and wrong, keep shoving the Doctor into said human societal definitions?
Moreover, I'm not convinced by any of the "explanations." They're just a fan who are trying to come up with in-world explanation based on basically nothing, rather than talk about why it matters, or doesn't, which is where they started from. Quite disappointing, with the "brain-washing to reduce suicide" Watsonian justification.
Anyway, all that aside.
Is the Doctor gendered? I think it is. Both because it was written by humans who wrote him as gendered, and because his actions are gendered. The whole show, the female companions, etc. Is it possible that the show-makers are going to move the doctor past being gendered? Or into being gender-queer, gender-fluid, or trans-gender (I put these apart on purpose)? Yes, it is quite possible. The show, like any show, is still a reflection of its time. But that would be change, as @techslut touched on.
Now, whence does the Doctor's gender come from? The body or the memories? That's something for the show-makers to decide, but if the Doctor were a human, I'd say memories would've definitely played some part in it.
I'm now going to share a personal anecdote. A decade ago I took medication that killed my libido entirely for about a year. I wasn't sexually attracted to anyone. At all.
And yet, I still found myself flirting with women, sometimes hitting on them, and preferring to be by them, all else being equal, than with men. What gives? I actually thought of it at the time and my conclusion was that I've already been conditioned, both by society and by my own physiology to behave in this manner, and even if you remove the physiological impetus, the behaviour it had conditioned in me remains.
Now, I want to raise a qualm with your definition of "cis" as "someone who doesn't care much about their gender." Some men and women who were born the gender they identify as care very strongly about it. Such as men who strive to be as masculine as they can be, or feminine for women. Are they not cis? They are, so the definition is lacking.
I also wonder if your definition of "gender" is good enough. Like the author of the video you linked, I think it might've been better to stay without a definition. Here too, I wonder, if men and women are treated the same, but say, they think differently, would such a world (hypothetical or otherwise) not have gender? Just sexes? And does your definition also address how people think of themselves?
Finally, I'd like to recommend you Tanith Lee's Don't Bite the Sun, a science-fiction book about a society where people can change their gender at will.
Because we can't think of things using their language, we have to use ours.
Well, ya, obviously. But if we try to imagine the doctor really existed, and humans are telling a story about the doctor, that's a more interesting perspective. Saying the intention of the author is above all is, well, kinda pointless.
Definitely. I have had a lot of self discovery on this subject when it came to sexual attraction. I've been condition to be attracted to women for the vast majority of my life. And while I can feel attracted to men, it takes particular circumstances and quite a bit of effort. I've never looked at a man and got an erection. Sexual attraction is an odd beast. If bisexuality was extremely common and accepted, would more people feel naturally comfortable to be bisexual, even if they, living in this world, are not?
That's not how I defined cis, it's how I defined "cis by default". Cis is anyone that identifies with the gender associated with their sex. Cis by default is anyone that is fine with the gender associated with their sex, but would be just as fine if they were assigned a different gender by society.
Then it wouldn't be gender, it would be two groups thinking differently. Like engineers and artists.
My bad, I didn't realize this is one term as a whole. And yes, it's a good distinction to make, I just hadn't come across the term before.
I was actually saying the opposite. That The Doctor can't help but be gendered, or told as gendered, because we wrote it. It might not be as interesting from a sci-fi perspective, but I think it's paramount to keep in mind while we discuss the cultural artifact that is the show. Though yes, your post straddles doing both.
History (ancient Greece, for instance), seems to show that this is the case. That bisexuality is also an acquired trait, to some degree. And that if it's culturally accepted to be bisexual, then more people are. I could very well be wrong here, but that's what I picked up.
I find it perplexing you think we can't help writing a gendered character. We have genderqueer people, and gender blind people, and agendered people, I'm sure they'll be fine writing such characters. And even if the author is gendered, it shouldn't matter, just like JRR Martin is able to write strong female characters, while being male.
Note that this is just based on my literary-socio-psychological observations, so it's quite possible we'll disagree here.
Look at Tolkienesque dwarves - they are not just people, they are humans.
Elves? Same.
Even the dragon, which is supposed to be quite different, can't help but end up as a human with quirks and traits, as he was written by a human.
Now, let's look at the concept of religion, or specifically, religiosity and atheism. Our current concepts of such in Israel are not only Western, but Christian for these terms. That caused an issue in Israel when immigrants from Arab countries were presented with terms couched in an alien framework.
Finally, I'd like to bring up the discussion we've had previously on conditioned sexual attraction in our own lives.
The analogy is this, that regardless of our own genders, the framework in which we see people and society is gendered. Trying to either a character and/or a society that is different, that is not gendered, is going to be really hard, and for a case of a show with multiple episodes and writers, all but impossible.
Is it possible as an idea, sure. Do I think most people who are already over 20 years old be able to do so? I don't think so. And I'm thinking of most sci-fi authors as well. "Gender" at this point is not just a trait of people, but one of the lenses we view the world through, regardless of how we apply the trait itself. And that'd require a cultural shift to shake off, and those take quite some time. Decades.
Edit: Realized it might be a bit of a semantic disconnect. When I say The Doctor will be "gendered", I don't necessarily mean either "man" or "woman," but that it'd be part of the gendered framework, and with gendered aspects.