I think these is some merit to the arguement that we should treat obese people (in the interest of full disclosure I'm usually in the overweight category by BMI) in the same way we treat smokers. Reduce temptation, avoid glamourising unhealthy foods, remove hidden sugers etc.
Fat shaming is like going up to each smoker you see on the street and screaming "eww that stinks" in their face.
Comparing yourself to gays is just #&#+$+
I'm not pro fat shaming, but spending time in East Asia and getting called fat publicly (even by close friends, but especially strangers) has led to lifestyle choices that have me down 50 lbs from when I first came. I saw that society here (in this unspecified country, not trying to make a blanket statement about East Asia) views fatness very differently, and that I needed to assimulate.
Being fat in the US, where I grew up, is viewed very differently and is normalized.
For example, in Canada I found myself accidentally ordering a large out of habit. What arrived was invariably more than I could eat. Easily one and a half times your daily calorie requirements in one meal.
In Spain I don't even remember a large being an option.
I think discussing this topic is quite difficult because there is no agreed upon definition of "fat shaming". I'm pretty sure that there are people who would count any measures to reduce obesity as fat shaming (i.e. "cancelling fat culture" or something ridiculous like that), and the discussion tends to get emotional very quickly.
I think we have fairly strong evidence that gay shaming doesn't make people less gay (there are homosexuals even in places where that carries the death penalty). I don't think we have equally strong evidence that telling people they'd look better and reduce their risk for cardiovascular disease if they lost weight doesn't help (at least some of them) to actually do it. At least I'm not aware of any and I believe the confounding variables with anything related to nutrition are so numerous that gathering solid data is extremely difficult.
There is also a problem on the low side of body weight. A lot of the models we see are unhealthily thin and/or are photoshopped to look that way. While a lot of attention has been given to how this affects body image for people who end up with eating disorders like anorexia, I wouldn't be at all surprised to find that the same images contribute to people feeling like they will never look like that and going the opposite direction.