This is another case of a foreign word don't have a good translation in English (and vise versa). Both 摆烂 and 让它腐烂 don't have the same tone as "let it rot".

To me, "let it rot" means watching something collapse with a sense of enjoyment. I cannot recall a Chinese word with this exact sentiment of the top of my head. But I can try to explain both Chinese words.

"让它腐烂" is the literal translation of "let it rot", word for word. It don't have the cultural and sentimental meaning behind it, merely stating the fact. More like "let the leave rot in the compost pile".

"摆烂" is probably what the article is referring to. Its meaning is similar to civil disobedience, and 躺平 ("lay flat", another word that was popular couple years ago).

"摆" means put, "烂" means something poorly made, broken, etc. "摆烂", together as a word, means "displaying a broken (bad) attitude, no matter the outside influence". However, "烂" also means rot, which is probably where the translation "let it rot" came from.

The original usage is much more playful, like your cat would lay on the floor no matter what toy or treat you give it, then it is 摆烂. But with the recent increase in pressure for many young people in China. 摆烂 and 躺平 (lay flat) become more of a act of civil disobedience and refusal to participate in the broken system/economy.

So 摆烂 is not a exact translation for "let it rot", but they do share the meaning of "no action" and the sentiment of joy. And "let it rot" sounds much cooler and concise than my explanation.