It's not just the idols alone though. It's the entire Korean entertainment / media / fashion / beauty industry. Idols and musicians are just one facet of the Hallyu cultural media complex, which is just as powerful in Korea as the military-industrial complex is in the U.S. (good luck trying to change American laws on guns in the face of all those political lobbies and pressure groups with vested interests...). Especially since there so much money to be made in the export of Korean beauty/fashion products and services, and entertainment products (drama / music / film / variety), many of whom do not want to face commercial and financial risks in going against what is perceived to be the market-approved standard for beauty.
There's also a related question of Kpop companies having the luxury / power / financial resources to uphold obligations to be responsible for societal influence. Capitalism, classism, and extreme competition are hallmarks of South Korean business culture. For many small and mid-tier Kpop companies, do they have the financial resources to survive if they try to promote idols who have healthy bodies? Can they guarantee their idols would succeed and gain enough influence to set the standard, if they do not match what the market desires? MAMAMOO can barely do it with their outstanding talent. Their predecessors in the 2nd-gen, the Brown-Eyed Girls, adopted plastic surgery in order to gain influence, despite being equally-talented vocally.
Of the Big 3 (or 4 if you include HYBE), JYPE is the one that focuses the most on promoting physical health in their corporate social responsibility projects. SM Ent does more education and community social services, YG Ent focuses more on charity donation and now apparently environmental sustainability. I believe HYBE is also moving towards more environmental sustainability initiatives in their CSR focus (and previously mental health). I don't know what Kakao Ent or the CJ group are doing. But in any case, that means that almost none of the major players in the Kpop industry are vested in promoting a healthy body image or lifestyle, except maybe one. And even then, they are less focused on the psychological elements of a healthy body and more on the nutritional and medical elements.