The Dark Brotherhood in Skyrim is supposed to be a contrast against The Dark Brotherhood in Oblivion. We’re supposed to cringe when we get to the sanctuary and Astrid basically says “No Night Mother, no black hand, no tenets, no rules”, we’re supposed to cringe when she either breaks or orders us to break the tenets. Lucien Lachance (who probably didn’t take the purification of his sanctuary lightly, considering he raised the two shadowscales) even says that sanctuary needs “a good purification”.
It doesn’t feel like the brotherhood, because it isn’t, it’s wrong, and it’s written like that on purpose.
Five Tenets, and how Astrid broke them/makes the Dovakiin break them
Tenet 1: “Never dishonor the Night Mother.” That’s mostly what Astrid does early storyline. The Dovakiin can even call her out on this.
Tenet 2: “Never betray the Dark Brotherhood or its secrets.” She sabotaged the emperor’s assassination by giving information to Maro (which led to the sanctuary being ransacked)
Tenet 3: “Never disobey or refuse to carry out an order from a Dark Brotherhood superior.” She directly refused to hear out a contract from the Night Mother.
Tenet 4: “Never steal the possessions of a Dark Brother or Dark Sister.” She orders you to rifle through Cicero’s journals.
Tenet 5: “Never kill a Dark Brother or Dark Sister.” She orders you to kill Cicero, who also happens to be the sacred Keeper of the Night Mother.
Doing any one of these things is said to invoke the wrath of Sithis, and Astrid committed them all, the ransacking of the circle the wrath of Sithis, her death being her redemption.
The Skyrim Dark Brotherhood questline is about returning the Brotherhood to it’s roots, saving it from the sorry state you find it in. It is a different kind of story than the questline in Oblivion, but I don’t think that makes it a bad story.