Everything makes perfect sense. But i find sort to have different culture meaning in english than załatwić in polish.

When someone sorts something out, he goes and solves something with available methods.

When someone goes "załatwić" something, it means he solves something that is impossible to solve or gets something that is impossible to get.

To sum it up, i think english vocabulary for sorting something out synonyms ends at some point english speakers can not pass. And than "załatwić" kicks in.