i think if you want a 3yr to be able to do things by theirself that you have to teach them at an earlier age, expect a bit of mess (with food, and no lid cups) etc
my 3 was dry during day at 2.6, though is still wet at night at 3.3 - but he sleeps soundly 12/13hrs
he can dress, feed, wipe his own bum, put shoes on and off, coat on and off, dress himself, undress etc - and has since he was about 2.5
i generally teach/allow children to try and eat unadied by 1ish, so by 2 they are usually capable of getting most food in mouths
same with cups, by 18/20mths i will let them have a lidless cup at table when eating, but a lid in bag/car if we are out
tbh most parents dont allow their child to learn/do it as it all takes much longer and gets messy, but that how kids learn
also think having a nanny makes a child more independant as again children often do for their nanny what they wont for their parents
therefore we know that they can dress, feed, wipe their bum etc yet often see the parents doing it for them