Do running belts actually stay in place without bouncing?

For most popular designs, yes, but only if you size them correctly. Across specialist subreddits like r/running and r/XXRunning, the most consistent no-bounce reports come from stretch-tube belts (FlipBelt) and snug spandex sleeves (SPIbelt, Sporteer). Loosely fitted adjustable-strap belts and overloaded belts are where bounce complaints concentrate.