Gender Dysphoria is a clinically defined condition. Here's a key thing, though: being trans does not necessarily imply dysphoria. And even when it does (I will admit, probably the majority of cases), the nature of the issue is that stigmatizing it as "abnormal" makes it worse. We can call it "feelings police" all we want, but the fact is that for someone suffering from a serious mental condition such as this, "hurting their feelings" can be a very big deal. Which may help answer your second question:
I have no issues with sort of playing into their psyche when dealing with them directly, that's what makes them the most comfortable, and that's what is recommended from the APA in terms of improvement of qualify of life.
However, it's how we interact with each other (when not interacting directly with them) that I find concerning... what I mean by that, is that pretending that "there's not a thing wrong" then also removes pressure from society on the scientific community to work toward a cure or more effective treatment options.
For example, if a person has schizophrenia, and hears a voice in their head named Eric...it's perfectly reasonable for us to maybe play into it and pretend that Eric is real for their sake in efforts to provide empathy and comfort while they're having an episode when directly dealing with them.
However, we would come here on CF and have a serious conversation about "whether or not Eric actually is real?".
We wouldn't be saying "well, who really knows was REAL is anyway...".
If people took the same attitude toward schizophrenia back in the day that they take toward Gender Dysphoria now, the drug Thorazine would've never been invented. The pressure on the scientific community to research that came from a sense of urgency from society that said "this is serious abnormality, we need to try to fix it".
From the time the first definition and categorization of Schizophrenia in 1911, within ~40 years scientists and researchers developed Thorazine, a drug that treats the condition with a 72% success rate. All in all...not too shabby.
Gender Dysphoria was first categorized and defined in 1980, what progress has been made by the medical field in the last 36 years? From my vantage point, all of the efforts have been centered around convincing everyone in society that it's not a condition, rather than trying to actually develop treatments for the condition. (aside from some clinical trials done in the UK with puberty-delaying drugs in the mid-2000's)
You're right in that the underlying issue is assimilation into society...when people are made to feel like outcasts or different from everyone else, it only exacerbates existing issues. However, at some point we went from looking at those scenarios and saying "wow, this problem really prevents a person from assimilating into society and causes a whole host of issues, we really need to figure out how to fix this problem!" - to - "we need everyone else in society to change to accommodate this one person's dysphoria-induced belief...calling it "a problem" makes them feel bad, so we all just need to pretend that there's absolutely nothing medically wrong here"
I think we can (and should) strive for a best of both worlds approach here. We can be sympathetic to the situation, without deluding ourselves (intentionally) into think "there's absolutely nothing wrong here". The fact that, even in societies where it's far more accepted, the suicide attempt rate still nears 40%, shows us that there's something wrong and other pitfalls that go along with this condition that make it worth of researching and seeking a cure. This isn't simply a benign issue that's strictly cosmetically centered where "if society just accepted it as normal, everything would be okay"