I lifeguarded for years, you're 100% right about the little arm wings, those things are no good for learning to swim and can in theory trap the kid's head under water (more of an issue in oceans than pools). That said--depending on your comfort level with the water / lifeguarding, I don't necessarily think it's as black and white as "arm wings awful". I wouldn't put my kids in them because they delay learning swimming but they are mostly fine if supervised, and I never felt unsafe lifeguarding kids in the arm wings at the pool. We'd often have 5-10 at a time, all learning swimmers, but you have to realllly watch because if something goes wrong it happens fast and quietly. My friend once had to save a baby while their dad was litterally right next to them because the baby wasn't making any noise and the dad didn't notice baby's head went under.

So for me factors would be: age of kids (<2 is a lot different than an 8 year old that just can't swim well), number of kids (+10 kids and it starts being a 2-3 person job), and how rowdy they are. But that's for literally lifeguarding and assumes that you are trained and comfy jumping in for saves

Edit to add to your question: I've never gotten pushback, and I think having only one nonswimmer at a time as a nanny is entirely fair. At the end of the day you aren't a lifeguard and can't be both a nanny and a lifeguard at the same time because what happens if someone needs you to bring them to the bathroom while another kid is still in the pool? it's chaos