A few years ago in a large brewery I'd worked in , a young brewer had decided on a bright idea of removing the zinc addition to boil kettle step , in order to try streamline the process in the brew house . About two weeks later the whole fermentation area was backed up with stuck fermentations, it caused a major incident, as he hadn't flagged anywhere that he'd done this , and it took a bit of investigation to discover what had gone wrong.
Tim O Rourke always used to mention at brewing lectures about German breweries not adding zinc in order to comply with the Reinheitsgebot, but instead use galvanised impellers on their wort pumps, with an interference fit between the impeller and the pump body.
My experience is that if you're intending to crop yeast, then you need to add a tiny amount zinc , I usually add it at the same time as I add kettle finings , dried yeast and liquid yeast sachets come with sufficient nutrients for one generation , but if you're intending to reuse yeast , a lack of zinc will impact on further generation health and performance . In particular you'd most likely see a fall off in attenuation , with final gravities creeping up over time , and some off flavours developing .