This is definitely true but a bit misleading. There are 3x the number of teams that have had top 3 picks since 2000 vs. teams that haven't had top 3 picks (I went back to 2000, because even though the salary cap wasn't instituted until 2006, top 3 picks from the 2000 draft would still hypothetically be in their primes/young enough/good enough to be star players for their teams post salary cap), so right there you're comparing the results of 24 teams vs the results of 7 teams - which obviously isn't a fair comparison. If your single criteria for having a successful rebuild/successful team/cup contender is the be all end all of winning a cup then yes, again I suppose your statement is correct. However, many of the teams without top 3 picks have been very successful and have been top level cup contenders (which is all anyone can reasonably hope for, unless you think Tampa Bay has been an unsuccessful franchise in recent history, or Washington was an example of a bad rebuild until last year, i'm guessing you don't think those things) even if they never quite pulled off the feat.
Since the 2000 draft these are the teams that have had top 3 picks (24 teams):
Minnesota
Ottawa - 1 SC loss
Pittsburgh - 3 SC wins, 1 SC loss
Washington - 1 SC win
Anaheim - 1 SC win
Carolina - 1 SC win
St. Louis
Chicago - 3 SC wins
Philadelphia - 1 SC loss
Phoenix/Arizona
Tampa Bay - 1 SC loss
LA - 2 SC wins
Atlanta/Winnipeg
NYI
Colorado
Florida
Boston - 1 SC win, 1 SC loss
Edmonton - 1 SC loss
Columbus
Montreal
Buffalo
Toronto
New Jersey - 1 SC loss
Dallas
And here are the teams that haven't had a top 3 pick in that timespan (7 teams):
Calgary
San Jose - 1 SC loss
Nashville - 1 SC loss
Vegas - 1 SC loss
Vancouver - 1 SC loss
NYR - 1 SC loss
Detroit - 1 SC win, 1 SC loss
So yes, it is true when someone says teams with top 3 picks have won more cups, and at a higher rate than teams without those picks.
However, looking at the teams that haven't had a top 3 pick and their continued success (for the most part, outside of Calgary) in the cap era. It's really hard for me to look at those teams and say something like wow "If only San Jose had that top 3 pick, they would have been able to win a cup over the last 10" or "Nashville lost in the finals because they were missing that top 3 pick." I just don't buy that line of thinking - considering how successful and how much sustained winning teams like Nashville, San Jose, NYR, , and Detroit (57% of the teams in the sample) have had in the post cap era -- even if not resulting in a cup (minus DET). For example, post cap, DET, SJ, NYR, and NSH are all in the top 10 in playoff wins.