I'm willing to believe that ... but do you have any numbers that actually support that contention?
I would define 'plays a major role' as either "played for" or "was traded for someone who played for" the team that (a) earned a top pick and (b) went on to win a title within a reasonable time frame.
From what I have looked at, I don't think that pattern has happened as often as you are suggesting.
I'm interested in the idea of undertaking a larger "case study" as far as this goes, but for now let's just take a look at a snapshot of the league this season.
Here are the top 10 teams in the league by record at the moment:
2014:
1. San Antonio
(Duncan - drafted #1, Leonard - draft-trade #15, Parker - drafted #28)
2. Indiana
(George - drafted #10, West - FA, Hibbert - drafted #17)
3. OKC
(Durant - drafted #2, Ibaka - drafted #24, Westbrook - drafted #4)
4. Miami (LeBron - FA, Bosh - FA, Wade - drafted #5)
5. LAC
(Griffin - drafted #1, Jordan - drafted #35, Paul - trade)
6. Houston
(Harden - trade, Howard - FA, Parsons - drafted 38th)
7. Portland
(Lillard - drafted #6, Aldridge - trade-draft #2, Matthews - FA)
8. Golden State
(Curry - drafted #7, Lee - FA, Bogut - trade)
9. Memphis
(Conley - drafted #4, Randolph - trade, Gasol - trade)
10. Dallas
(Nowitzki - drafted #9, Calderon - FA, Ellis - FA)
Sometime in the next few days, I'll make a post that goes a little bit more in depth into how these teams were built, and also look at the top 10 teams from the past couple of decades. I think this group is probably pretty representative. There are some exceptions, but pretty much every team is relying on at least one player they took in the top 5-10 of the draft.
Note:
Why look at the top 10? History tells us that one of these teams is likely to win the title this season. Generally speaking, the team that wins a title finishes with a top 5-10 record. We could also look at top 5 in offensive and defensive efficiency, because those tend to be pretty good indicators of which teams are truly elite and have a shot to go deep in the playoffs. For my purposes, overall record is good enough to give us an informative sample of the "top tier" of teams.