You can also assign it to the triple click shortcut in Accessibility. You probably can to the double/triple back taps too, though I haven’t tried.

I do use a standalone Lectrofan for sleep as I prefer my noise machine to be across the room and Alexa-controlled (via a smart switch), plus it’s louder and the brown noise is “browner.”

But I keep iOS BG sound mapped to the triple-click shortcut for when noise-cancelling just isn’t enough in loud restaurants etc. It works great with AirPods for reducing my noise sensitivity issues.