I've never seen or heard that it harms anything with extended use. Please give details about your shotgun. In the case of the gold-colored coating on pipe bands, both examples I know of involved the same model - a Peterson 314 that was a tobacco coupon giveaway back in the 1980's. I suspect the coloring was extremely light and weak, and possibly could have been scrubbed off with a magic eraser; however, because of that experience I wouldn't use Flitz (or any other polish) on a plated item.
As I stated, I've only used it once on each item;however, I've used it heavily on stubborn silver pipe bands before, and saw no degradation at all of hallmarks, while still removing nasty spots.
But I don't know what problem you had with your shotgun - maybe it was such that certain warnings need to be stated. Did it have to do with blueing? I've used Flitz on blued S&W revolvers with no problems, and we know that the blueing rubs off of the sharp edges eventually, just by hand contact.
Sorry for being so verbose - I just hate to see people avoid certain products or techniques because of generalizations on forums that I haven't found to be entirely accurate.