Yes, but by the time of Bonnie Prince Charlie the Stuarts were hardly very Scottish. I don't dispute their distant ancestry of more than a century before Culloden. They weren't very Scottish by the time of Charles I, after the Restoration.

James I of England got to be king for the simple reason that Elizabeth was The Virgin Queen. Hence the son of Mary Queen of Scots got to be the English king too. I can't remember how the Tudors got hooked up with the Stuarts so as to have a common heir.