No engineer but I'm guessing that frictional loses from a rubber belt will be way higher than a greased, bushed (are cycle chains bushed?) chain. Also, plastics and rubbers are elastic. That's their unique benefit in a lot of applications. In fixed gear to gear power transmission, I'm not seeing a stretchable material as a benefit. Off the top of my head I can see a need for tension adjustment, either by idler gear or movable gear, and the load on the gears' arbors would be increased since they'd need to be spaced to maintain tension on the elastic rather than just a fixed distance that suits something less elastic like a steel chain.
Drive belts of umpteen varieties have been around for 100+ years but nobody has thought of using them on bikes until now? Seems unlikely. Sounds like a marketing gimmick to me.
Yes, they use them on cars. Idk. Maybe if a metal chain failed in a car it would cause catastrophic damage. Maybe the belts are cheaper or are more forgiving to fit to fixed position pulleys/sprockets/idle geared assemblies.
Proven technology is proven. There's no such thing as newly sliced bread on something as basic as this imo. Get yourself a grease guard and don't dress like a goddamn Jehova's Witness if you're fussed about dirtying your Sunday best.