That was the story we were told too, but I later found out the Bruces in the US apparently came from Robert’s grandfather in France, not directly from Robert. They arrived in the colonies early as Huguenot Brousse... later changed the name to Bruce.
My family trees shows me descended directly from him through his daughter Matilda to Robert the II, III, to John Stewart I, Lord of Atholl, and down through his daughter, etc. I made an error when I said I was related to Edward I through his daughter. The line actually goes from Henry II (haven’t gone any further back yet) down through John I (Magna Cartha), down through Henry III, Edward I, Edward II, Edward III, Thomas Plantagenet of Woodstock, 1st Duke of Gloucester, and then down through to my mother. They’re both on my mother’s side of the family. I take it with a grain a salt, because the DNA only goes back to the 1700’s.