This is not a good analysis for many of the reasons that have been suggested already. You don't get to call something a fluke just because it doesn't conform to your theory, and you also don't get to arbitrarily set the sample to the time in which your theory is confirmed. From 2006 to 2012, we had a new champion every single year. And you obviously don't get to determine parity by one measurement or datapoint (Stanley Cup winners) that confirms what you want it to.
You've started with a conclusion and then backtracked to include only the observations that confirmed your conclusion, then you suggest the evidence is conclusive. That's just bad analysis.
Not to mention the fact that you ignore the fact that there is no shortage of examples of teams who collect top picks and never get competitive. Yes, Chicago had Toews and Kane... they also had hundreds of other factors that cumulatively delivered them sustained success. Yes, LA had Doughty, but they also had an uncountable number of factors that contributed to their success. And, hell, Pittsburgh was a giant disappointment a little over a year ago for failing to sustain their success, despite having Crosby and Malkin that whole time. The narrative that teams can't succeed without lottery talent is nonsense. Lottery talent is obviously a big help, but it's not a prerequisite.
Additionally, something that people overlook when evaluating the success of the Hawks and Kings: they both drafted more often than the rest of the league. Chicago, for example, from 2001 to 2010 made 102 selections at the draft, far more than the rest of the league during that time-- the next closest was 8 picks fewer. That's more than a full draft class more than the second most frequent drafting team! Chicago's success on a per pick basis at the draft was not particularly impressive. They did not have success at the draft because they identified talent; they had success at the draft because they had so many damn picks. They had Toews and Kane, but they also had a nonstop flow of solid roster players as a result of sheer volume of picks. In a cap world, this is just as important as having lottery talent. AFAIC, it's more important.
The "rut of mediocrity" that gets perpetuated around here is the biggest myth around. If you're a mediocre team, you don't need to tank or go for a full-blown rebuild. You need to stop trading picks and you need to add one, two, or three extra picks per draft, even just in the mid rounds. Within ~three years, you'll yield NHL talent that will help you turn your team around even if you never pick in the top five, provided your management is astute in other areas as well.