I can see the positive & negative side of it though. However sheltering the youth from all negative feedback is not a good thing either. People need to grow up facing adversity.
Feelings are great, but when you put them in the driver's seat you don't get the chance to see how they can change and how you can grow in the process. It's the difference between being scared of flying all your life and getting past that fear. There's pride in overcoming your fears. You feel less like a victim.
For example, people in India are generally opposite of what you might call 'precious'. Parents will very openly call out their kids on their body size, their appearance, there whatever. Young men and womanly will openly (and sometimes, far too much) discuss relative attractiveness of persons -- often admitting that while they themselves are not that attractive, so and so is even uglier, etcetera. They are, I would imagine, not in any sense of the word 'precious' or coddled. The South Asian cultures are also the epitome of what might be considered 'collective' culture, versus the more American individualistic, 'me-me-me' culture.
And yet. People never ever ever get naked. Ever. Not even when they shower. In their private bathrooms. With no one watching. There's joke about how South Asians don't get naked even when having sex (which, I'm told, appears to reflect reality). You would imagine, with people so comfortable with friends and families (note South Asian men have traditionally been verrry comfortable being physically close to other men, including holding hands, frequent hugs, etcetera), they would not be as shy about nudity. And yet they are.
Here's another datapoint: the Japanese, who are still culturally very family-community centered, will get naked at a drop of a feather.
Use your 'precious generation doesn't want to get naked' theory to explain these, and your theory will pass the basic test.
Comparing Apples to Oranges is exactly what's needed here. By doing so, I'm helping isolate the variable OP suggests affects Apples is not so. Imagine he'd claimed that Apples are suddenly brown because of 'those damn young trees'. If I show that those darned young trees elsewhere don't have the same issue, then there'd be a greater need on the OP's side to actually provide the proof that it's those darned precious young people.
OP needs to provide a greater proof that it's because of those 'precious youth'. I might argue that it's due to increased cultural influence of more body-conservative cultures, or because (ridiculous as it may sound) online nudity is now so common, people don't want their online versions to be identified to offline bodies [which is VERY different from what OP argues].
Except its not a "very short term shift that defies their parent culture", its a long term drift in the broader culture, as the article notes: Gen-Xers are notably different than Boomers on this, in the same direction (though not to the same degree) as Millenials.
The perceived Japanese sense of modesty and prudeness is a relative new phenomenon, instigated after WW2 by American prudeness.
Over here in Belgium, such prudeness is pretty rare.
There's an air of...entitlement(?) to comments like this. Nobody is forced to be naked in front of others, and a gym isn't a public accommodation. If it doesn't work for you, don't go there.
Gym membership is not a "right". Everyone's particular sensibilities can not be taken into account. If the market demands privacy the market will deliver privacy.
By the way, a gym is absolutely a public accommodation, but it's true that gyms for adults generally do not require you to get naked in front of other members of the gym (you're free to leave sweaty). Still I can't think of a good argument as to why a gym should go out of its way to avoid allowing its members privacy in the locker room...