But, if I'm not mistaken, the damage you speak of is caused by disrupting sleep. So the idea is that if a white noise generator can drown out the interruptions, it helps you sleep better. There are studies that say noise doesn't help, and there are those that say it does. And it's all real-life medical research, so pretty unreliable. I found one controlled study (https://doi.org/10.1093/sleep/zsag001), but it's small, and only 7 days, and it's in a lab, so not guaranteed to translate to normal conditions either.

And don't ignore the placebo effect. Perhaps believing that pink noise helps is sufficient for some.

Perhaps a better recommendation is good ear plugs.