The claim about skill use is generally true. But the bigger issue is addressed in the 4e PHB (p 258):
Sometimes a quest is spelled out for you at the start of an adventure. The town mayor might implore you to find the goblin raiders’ lair, or the priest of Pelor might relate the history of the Adamantine Scepter, before sending you on your quest. . . .
You can also, with your DM’s approval, create a quest for your character. Such a quest can tie into your character’s background. For instance, perhaps your mother is the person whose remains lie in the Fortress of the Iron Ring. Quests can also relate to individual goals, such as a ranger searching for a magic bow to wield. Individual quests give you a stake in a campaign’s unfolding story and give your DM ingredients to help develop that story.
And from the 4e DMG (p 103):
Player-Designed Quests
You should allow and even encourage players to come up with their own quests that are tied to their individual goals or specific circumstances in the adventure. Evaluate the proposed quest and assign it a level. Remember to say yes as often as possible!
This clearly contemplates that the player will introduce goals for his/her PC, including facts about parents and fortresses and the like. But it is closer to Dungeon World - informal principles ("say yes") and techniques (conversation between player and GM) - than to Burning Wheel's formal framework of skill checks.