I think it'd be almost doable if the vest were as long or longer than the jacket.
This is one of the jarring points. The padded gilet only reaches the waist.
Over a jacket or under?
A vest under a jacket is common. Over I've never seen. A vest under a jacket is common enough you'll see jackets sold with a zip out lining.
Over. Although I have seen the giacca con gilet as well. Have to get a whole extra coat for that.
Perhaps the general horror at the idea, is that outside Italy, this is not done.