"In terms of sex, yeah sure - but we don't need to talk about the concept of gender at all to do this"
"We should be parenting our children as individuals on a case-by-case basis"
Unfortunately, this isn't as easy as you would think. Child raising gives limited windows of opportunity and you don't have much room to find out. Sometimes by the time you find out it's too late so you have to take a gamble on some things and any reasonable parent would go by the actually proportions, not by artificially imagined equal proportions. The thing is, I tend to agree with this principle but the fact or reality is you can only pursue it opportunistically, you can't impose your will over reality. There is no magic bullet here. You can't actually entirely make the separation you're trying to make. There are improvements that can be gradually made here but if you're just going to start meddling based on principle alone you're only going to cause more damage.
There's also a very careful line here that you need to think about. You're talking about binary gender being a social construct being imposed but so is gender neutrality and what is being discussed here often equated to imposing that. Get them why they are young. It's a form of indoctrination that falls into the extreme and threatens to turn millions of children into a socialist nihilistic experiment. The horrifying thing is that with the malleability of the human brain and its ability to ignore or override nature this is actually possible. You may think that is a good thing but then take religion as an example. You are proposing to replace one moderate yet beneficial evil with another more uncertain evil. On the one hand many may have surprisingly reasonable views but on the other I think you'll find not so many such people finding it reasonable to solve a problem with the naive wholesale reprogramming of children.
In my experience I find that most men and women overall don't have a problem with a gender split with a moderate identity for each gender unless you make it out to sound like they will be socially unacceptable if they don't. I find that many will have complaints about certain aspects and suggested improvements. Those I find strongly against it (for an extreme position on gender neutrality and separation from sex) seem to be following dogma, have not really thought about it, are not immediately affected by it, have vested interests (being outliers) or a huge chip on their shoulders being unhappy with life in general. For the most part stupid aspects of a gender split have been eliminated from our society.
There is a funny question you have to consider as well which is about hijacking identity. What if biological men and women voluntarily want to be distinct? Lets say that you establish trousers for all, skirts for all, etc (something I support for many segments of society). They no longer have a way to be distinct so invent a new method such as a new unique set of clothing for each not seen in any other culture. If you take that then it's not really cool and they have to invent yet another protocol. At that point you have to ask is this really separate and who is it that's being oppressed or suppressed? Just as most wouldn't dress in Islamic attire even if they weren't Islamic. Technically speaking someone can but you just usually wouldn't out of courtesy and so not to give the wrong impression. Also if you're going down this route, what about religions and other cultures with these identity related behaviours and traditions? What about people raising their children to be religious with far more "social constructs" than in our society? It's hard to say where it is but there is a line you have to draw here somewhere. So we're secular and have choice but there is a limit after which you impose on parental and personal freedoms as well as the right to a particular identity.
You have a part of this argument where sure, people are treated unfairly or inappropriately and there's no reason for that, it puts no one out for that not to happen. There's another part of this argument however where you can put people out, a lot of people out, simply for the unusual circumstances, conflicting ideals, or excess demands of a few. Extreme caution needs to be taken here. If people cross that line it can genuinely cause a falling out between two groups of people with a real sense of contention over identity rights. The last thing those who mean well and want to further advance the cause of people getting along want to do is to pit people against each other.
Problems with gender separation are largely best left to cure themselves without taking radical measures especially when those typically come from an extremist section of a minority portion of the population. Unfortunately not every group not matter how small can receive special or sometimes even preferential treatment.
"But once we recognise that the traditional concept of gender isn't 'correct'"
You're never going to find anything entirely correct. There are rarely the best of all world solutions that are superior in all aspects. I've had to confront this many times in designing algorithms. There's always a scenario with some approaches that isn't going to be best catered to. Sure, a selection of trans people or anyone in any segment might not feel entirely catered to and put out that treated like royalty but the fact is that so far there's no indication that the sheer majority of people, even trans, aren't reasonably catered to under the current system which itself is evolving and improving. If I had to optimise for a workload of 90% reads I would not be expected to optimise an algorithm to ensure same performance expectations for reads and writes at the heavy expense of all the reads and the overall performance.
"it's when people expect everyone (or anyone) to adhere to that traditional concept"
This is usually fine but potentially one of the problem areas. Sometimes you have to put the individual before the group. You can't have everyone put themselves out because of one self centred person. However sometimes the group can take it too far as well. I actually find that throughout much of society that isn't a problem but there are specific segments where it is. Normally, as soon as someone exposes that they are not playing the game, the response is simply, oh sorry or something fairly benign. Mistakes will happen, people will make assumptions, you don't demand they never do, you don't be a horrible person and give the wrong impression, not announce things, not give disclaimers just to be able to get all grouchy about people having the wrong impression and when people do because those things happen you correct it in a civil manner. A great disturbing trend among society is to hold people to too high a standard of infallibility. We should have more norms and philosophies that assume that people will get things wrong. When things don't go down acceptably, you deal with that there and then. Those kind of extreme problems have to be solved tactically, not strategically.