The ability to do those things, the ability to turn up nuggets of gold in unlikely methods in addition to drafting at the top of the draft is how teams speed up that process. 7 years may be a fair benchmark for a rebuilding team reaching their full potential but a base built by superstars at the top of the draft and augmented in other ways does not take 7 years to become an entertaining playoff team or even a contender.
For sure, the Av's turned it around in a big hurry via the draft: MacKinnon (1), Landeskog (2), Duchene (3), O'Reilly (33) and of course Stastny (44).
From 39 points in 48 games last year to 112 points in 82 games this year, its one of the most remarkable turnarounds ever.
So I think in some ways you are cherry picking the teams in your examples as having to have won the Cup in recent years. Plenty of teams have improved after having done well in the draft for successive years but just haven't put it all together for the ultimate prize.
As you rightly point out it doesn't have to be a top 10 pick to make a huge impact. Look at Palat (208) leading Tampa in scoring this year.
I would say Pittsburgh, Colorado, Tampa, Chicago (today's Chicago, not necessarily the Thomas-era Chicago) and Anaheim can say their level of success is directly due to great successive draft choices.