It’s absurd. Idols are highly skilled & trained individuals who have been preparing half their life for the vocation they are doing.
No one told Lebron or Messi when they were young, “please get some rest and take care of yourself. I’m worried.”
This kind of thing is usually focused on the girls.
The difference between the two comparisons is the sports industry has multiple levels of regulations,laws,rules etc and western culture prioritizes individual above all else. The total opposite exists in most of Asia, and the idol industry has gained a reputation of having slave contracts and very shit agencies that misuses and abuses their talents from the old sm entertainment to all the spire and hook entertainments out there. Plus the toxic cultural mindset of hard work means you are doing well, that saturated our society where you'd have people work themselves to death or injuries that are entirely avoidable, there's been countless documented cases of idols working with a broken leg in a cast etc when they should be resting among other things like that. THAT'S where this trend comes from where fans tell them to rest, it didn't stem from some toxic controlling mentality it originally came from geninue concern about their physical and mental well-being.