I'm home from work now, so I have some time to provide the analysis you've asked me for. This is not meant to be all-inclusive, but rather just the things that jump out at me right away.
First of all, the idea that this is a speech to be delivered is a problem. To whom is it to be delivered, and on what occasion? Without a setting in which the speech is necessary, it's really just a long exercise in self-justification. I get that trans people face the need to justify themselves on a regular basis (a woefully too-regular basis, really), but I'm not sure it's the best choice here. You see, this leads me to the main issue which is that in the Star Trek setting, it's probably unlikely that the self-justification speech would even have to be delivered in the first place.
For starters, the Star Trek setting in general is broadly a tolerant one, where people have set aside their petty differences and mostly work together, at least within the worlds of the Federation. Given the advances that trans people have made in just a handful of decades in the 20th Century and the first decade of the 21st, it seems silly to assume that trans people are a burning issue anywhere in the Federation by the time of the Star Trek setting. Gene Roddenberry deliberately made his bridge crew as multi-cultural as possible within the constraints of 1960's television production. He envisioned a world where all the little differences between people mattered not at all, but where the most important distinction was the content of their character.
On Betazed specifically, it seems that the issue of trans people should be especially a non-starter. They are a race of telepathic empaths. If anything, they would be showing the less-emotionally developed races of the Federation how it's done, not the other way around. An entire species of people who are constantly skimming the emotions of the people around them would certainly be far more understanding of trans people. For that matter, in a species of telepathic empaths, how could any minority group ever even be closeted in the first place?
This runs headlong against some of the autobiographical information contained in the story. For example, for an emotionally mature species of empathic telepaths, why would a trans person being a relative of a member of the ruling caste be an issue at all, let alone one that would "bring strife to an otherwise peaceful society?" The Betazed are supposed to be the mature empaths. There's no room for ignorant bigotry in such a society, and along the way to creating their peaceful society, such ignorance and bigotry would have been stamped out in the distant past.
Enough of that. I could write for days about the above alone, but there's a few other things I'd like to touch on as well.
The use of 20th Century science fiction as a symbol bothers me. However, I'll give you a pass on it because you're not doing anything that the entire rest of the collected body of authors of the entire Star Trek universe doesn't do too. To wit, why is the 20th Century so fascinating to the people in the Star Trek setting? With all of history to choose from, why does Star Trek lore always come back to the 20th Century when they want to show their future people being interested in their past? It's because that's when the people who are writing the stories are alive. We want to think that we're awesome, so much so that the people of the future will find us fascinating.
Also, there's the problem that the art, literature, cinema, and music of the 21st, 22nd, and 23rd Centuries have not been created yet! We'd have to use our puny turn-of-the-millennium brains to imagine the art of the near future so that the people of the distant future could be nostalgic for it. (And if there's one thing that we've found again and again, it is that any serious attempt to extrapolate the art of the future, even just the future a few decades away, is almost always hilariously wrong. In the 1930's they thought everything in the 1950's would be streamlined like the fastest trains, and driven by propellers and pneumatic tubes. In the 1950's they thought everything in the 1980's would have rockets and fins on it. In the 1980's they thought everything in the early 2000's would be day-glow colored and made of spandex and vinyl.* And so on.)
Speaking of the future, does anyone have surgery anymore? The impression I get from Star Trek is that almost every major health issue, particularly non-emergencies (like having a Nausican stick a knife through your heart) can be fixed with a wave of a magic medical tech device. I'm not sure this would be the pivotal event in the life of a Star Trek person's life the way it is in modern times.
And that leads me to this: I really think the whole story is a literal projection of a very specific modern cultural issue into a future setting where it simply doesn't fit. When Roddenberry wanted to talk about race in the Star Trek setting, he didn't emerge from the writer's room with a script about a planet where white people and black people still wanted to mess each other's shit up just for being white and black. He had a mixed-race bridge on the Enterprise, after all. So if a ship of the Federation was integrated and it was no big deal, why would it be a big deal on a member world of the Federation? Instead he made the famous people who are black on one side and white on the other, and their counterparts who are white on one side and black on the other. Surely there is some kind of allegory for being transsexual that could be projected into the Star Trek setting so that it fits better.
I've gone on overlong, so I just want to address one more thing. The author insertion character being a member of the ruling family of an entire planet, the same character being close enough to a canon character (Riker) to be a sort of protege', the bit about Ambassador Troi having a hand in the author character taking on the name of her very own daughter-- it's all very Mary Sue. That's all I want to say about that.
* I'm still sad that this didn't happen.