Okay, I object to the thought that Harrier DuBois could intuit the mechanics of the death note.
He WOULD solve the case though.
Harrier's Primary skill as a detective isn't deductive reasoning, it's his ability to say batshit things at people until their brains shut down and the truth slips out.
Light would present Harry with a perfect Alibi, a story that makes the very idea that he is Kira sound COMPLETELY ABSURD, and it would be completely ineffective because Harry doesn't want to ask him about Kira. Harry wants to ask Light if he can have some money, and also about his favorite brand of potato chips. This inevitably leads to Light going on a fascist rant about purging the scum of society, during which time Harry has started looking through Light's trash for glass bottles and found the mini-TV stuffed in the potato chip bag.