Yeah, and skaters (including my DD) have sprained ankles and broken bones doing off-ice in sneakers, too. Anything can happen, but as I said, I think athletic shoes are a must for off-ice.
I remind my students that they need sneakers for off-ice class, so they're usually prepared. However, some skaters show up in clogs, flip-flops, sandals, boots, heels, etc. Better for them to take off the footwear than jump in 99¢ thongs. (the flip-flops, not the underwear, lol)
Obviously, these were the rookies working on singles. The skaters working on doubles and triples always show up prepared with socks & athletic shoes, even if they wore Uggs to the rink that day. I wasn't teaching the classes nor were they my students. I think the off-ice instructor should have had them them miss the class or borrow sneakers, but it wasn't my call. IMO, either alternative would be a good lesson in showing up prepared. "Well, you won't do that again, will you?" "I guess we learned something today!" go a long way in building responsibility.
Barefooted skaters used the ball of their foot to similate the toe-in. Not correct, but impossible to repeat with a boot & blade on the ice. Beginners are just learning which foot to use in what order and how to transfer weight and vault, so it's not necessary for them to break a toe for a Toe Loop, although it gives new meaning to the name, lol.
I don't know about the "use your sneaker to simulate the edge for lutz/flip." At the higher levels, after they've mastered the single versions, they can probably pull that off, but it's really difficult for kids to understand why there's a difference and hold their foot in that way, which can also turn an ankle if they lose their balance.
I'm going to try the basketball sneakers and see how they feel. Thanks for the tip, Doubletoe.