Both brown noise and shoegaze remind me a lot of the Safe and Sound Protocol by Porges. It's a therapeutic intervention designed to increase cues of safety by filtering common music into the range of human speech. The creator of the protocol has done some research demonstrating efficacy and come up with a plausible theory for why his protocol works[1]. Maybe his work could add color as to why shoegaze and brown noise is so impactful for some people? Not really sure, but seems related.

[1] https://www.emdrwestonsupermare.co.uk/wp-content/uploads/201...