The conspiracy isn't that New Orleans won the pick to help the league because the league owns the team. The conspiracy is that the league was trying to get the highest price they could, and if they threw in the 1st overall pick when a true franchise player was available, then they could not only attract a buyer but get a better price.
Nobody believes that the league fixes EVERY draft lottery and EVERY playoff series/championship. But there are at least two cases (Lakers vs Kings 2002, Heat vs Mavs 2006) where the officiating was so ridiculously one-sided that the LEAST ridiculous explanation is that it was rigged. And with one ref (Donaghy) coming forward to both confirm it and also to give the mechanism and the main actors involved, the conspiracy whispers grow.
The NBA is so centrally-driven with Stern insisting on such dictator-like control, it's easy for people to start thinking things are rigged. Especially when so many improbable "storylines" (as one disgruntled team exec said after the Cavs won last year after losing Lebron and with their owner bringing his sick kid to the draft) seem to happen.
I do think the NBA has used league directives and ref assignments to influence the outcome of some playoff games. I don't think that they rig the draft lottery. however, there have so many "coincidences" in the past decade (I count 6), that if we ever find out that it was indeed rigged I would not bat an eyelid in surprise.