That's an interesting question and I hope it gets answered. About the only semi-expensive jewelry piece I own is my Omega watch with a stainless steel band and they recommend good old soap and water and I scrub the links with a toothbrush.
vinegar is an acid that painters regularly use to 'etch' the surface of smooth metals so they take paint better. I don't see that as being any good for a shiny watch band. To prepare metal it is applied and left to sit for a few minutes, maybe you could get away with a briefer application.
Makes a good household/bathroom cleaner/disinfectant too. A little spritz o' vinegar a little spritz o' hydo. peroxide, and you're ready for surgery..
I use an ultrasonic cleaner with plain water and a bit of dishwashing detergent; while it may not have the polishing action you may be looking for, it does a nice job getting all the buildup from the cracks and crevices that are inaccessible. I'd remove the band from the watch first though, because you probably don't want to subject the watch itself to the vibrations.
I didn't know you're supposed to clean watch bands. I have a Speidel that has to be more than 35 years old. I'd be afraid to clean it now
It looks new to me. It's apparently all stainless steel and was made pre-etched. Actually cross-hatching on each link.
Well I just noticed if I stretch the links apart I can see 35 years of accumulated crud. An archaeologist would have a field day. I'd better leave it alone. It's probably a crime to disturb an archeological dig without a license.