The only time I've seen "Someone quietly left" was when I was using the Nook stop and a guest left on their own by pressing the "-"
A message popped up asking me to close the window, but I ignored it and next thing I know it says "Someone quietly left"
It did not save but neither of us lost our progress or what was done on the island
If your guest leaves through the airport, your gate will stay open. If you press "-" to kick them out, your gate will close and they'll be forced to leave. And if they press "-", it's the same as them using the airport. Your gate stays open and the game saves