Whether a fat person, or any person meets the qualifications of being unhealthy is a matter of empirical verifiable fact, so this analogy is useless.
And you are treating being physically unhealthy as being unhealthy while not doing the same in regards to mental health. This is what is absurd. We know that many studies in regards to affirmation therapies and treatments are flawed because they lack proper control groups, and we see when control groups are offered that they aren't working.
Let's go to reassignment surgeries as a quick example (as I have that information more readily available).
Not long ago there was a large Swedish study on the mental health benefits of getting surgery, one of the largest out there. You probably heard news reporting on it as it went viral claiming that surgeries were beneficial... the correction to that study was basically not picked up by the media at all.
It was called out and a correction had to be offered.
Why? Because it didn't bother to use its readily available control group.
Turns out that those that didn't get surgery had the same exact trend, which caused them to have to say in the correction that there was no advantage of getting surgery.
It actually gets worse, however. Let's look at more data that only came out once the author was called out (that they themselves then published),
Who had signs of any sort of mood disorder?
Those that had surgery: 98
Those without: 88
Who had signs of any sort of anxiety disorder?
Those that had surgery: 85
Those without: 62
Who was on antidepressant treatment of some kind?
Those that had surgery: 301
Those without: 292
Who was on anxiety treatment of some kind?
Those that had surgery: 215
Those without: 149
Who was hospitalized following a suicide attempt?
Those that had surgery: 13
Those without: 7
So, not only did they have to come out with a correction that surgery doesn't give an advantage, they had to also come out and say that it gave rise to greater anxiety.
They also said that the other categories have numbers too close to tell for sure, but I think it is important to note that in every one of those "too close to tell" categories that the surgery category is worse. What that indicates is that either there is only negligible difference in those categories (and that coincidentally the surgery category was all on one side) or that surgery also makes it worse (like with anxiety).
So, world famous study that went viral in the news and was used by trans-right activists turns out to have had such a bad methodology that after being called out and corrected shows the exact opposite. This is what happens when you use control groups with these types of studies. Makes you wonder why it is that pretty much every single study around gender-affirmation lacks proper control groups and what the result would be if these studies were actually done properly (especially when every other dysphoria doesn't default to taking the affirming route).
So, if we can see the general trend that doing something will worsen your mental health, why shouldn't we call it out in the same way as things that will worsen your physical health?