Narcissa also made Snape promise to do the job if Malfoy failed.
I reread that chapter and you are correct. There is still something else bothering me, though: after what Dumbledore had shown Harry about Tom Riddle, it should have been obvious that TR/LV might be the Half-Blood Prince, and it would have been clear that Harry should ask Dumbledore about it. To be sure, for the potions book to have actually been LV's would have been too much like a replay of HPCoS, but it would have been logical for Harry to ask. This would likely have required some logical means by which Dumbledore could fail to figure out who the HBP was, but would have absolved Harry of the apparent blame for Dumbledore's death.
BTW, one thing I've disliked about the last three books is that Harry Potter's decision keep getting people killed. To be sure, he may not be morally to blame but nonetheless he's not totally innocent either. Worse, key plot aspects of LV's plans resolve around Harry such that if he simply did nothing, LV's plans wouldn't succeed. IMHO, it's a serious dramatic flaw when the world would be just as well off without the hero of the story.