Since 2000, 5 times a team has won Cup within 5 years after losing in the SCF. Only twice during that same period has a team won a Cup within 5 years of having 1st overall pick. So it certainly does seem that making long playoff run and falling short is better path towards winning a Cup than being crappy and getting top pick is.
I think the arbitrary window you looked at is a little flawed. Teams who draft #1 overall or even in the top 5 are there for a reason, and usually many reasons, so even getting a franchise talent it usually takes years to really turn it around.
Since the lockout the only teams to win without a player on the roster that the winner drafted in the top 4 overall are the 2007 Ducks, who won on the back of signing Niedermeyer and Pronger, two top 5 picks, in free agency (and Selanne picked at #10), and the 2008 Red Wings, whose key contributors Lidstrom, Zetterbergy, and Datsyuk were all drafted in later rounds. The Bruins in 2011 kind of count since Seguin wasn't a key contributor in his first year and I believe they acquired that pick through trade.
Even if you take it back to another arbitrary endpoint, 2000, when the lack of the cap fueled superteams like the Avalanche and Red Wings, many of those teams were still captained by a top 4 pick the team drafted at some point.
2000 - Devils - Niedermeyer 3rd overall '91
2001 - Avalanche -
2002 - Red Wings - Yzerman 4th overall '83
2003 - Devils again
2004 - Lightning - Lecavalier 1st overall '98
Lockout and Salary Cap
2006 - Canes - Staal 2nd overall '03
2007 - Ducks - Selanne 10th overall '88 (Pronger and both Nieds were top 5 picks for other teams)
2008 - Red Wings (Lidstrom, Zetterberg, and Datsyuk were all out of the first round)
2009 - Penguins - Crosby 1st overall '05, Malkin 2nd overall '04, Fleury 1st overall '03 (4/5/6 years after their picks)
2010 - Hawks - Kane 1st overall '07, Toews 3rd overall '06 (4/5 years after their picks)
2011 - Bruins - Seguin 2nd overall '10
2012 - Kings - Doughty 2nd overall '08, Kopitar 11th overall '05
2013 - Hawks again
2014 - Kings again
2015 - Hawks again
2016 - Penguins again (10+ years after drafting Crosby and Malkin, but they were still a Cup winning core)
2017 - Penguins again
2018 - Capitals - Ovechkin 1st overall '04, Backstrom 4th overall '06
2019 - Blues - Pietrangelo 4th overall '08 (and 1st overall in '06... but lets not talk about that)
2020 - Lightning, Stamkos 1st overall '08, Hedman 2nd overall '09 (Took over 10 years for them to add enough to their top 2 picks to win)
2021 - Lightning again
2022 - Avalanche - MacKinnon 1st overall '13, Rantanen 10th overall '15, Makar 4th overall '17 (9 years after drafting MacKinnon and they had to add another top 4 talent and wait 5 more years to win)
It's incredibly hard to build a winner without at least some bit of talent gained by those top 5 picks. In the salary cap era, simply having these types of talent to build a team around, even if they have an expensive cap hit, is pretty clearly an advantage in roster construction. And almost all of these guys are staying with their original teams, both on a contract buying out some combination of RFA and UFA years and then even after they'd normally hit UFA.
It seems like many of these teams are strongest in that window 5/6/7 years after drafting that key player, likely when they are on their second deal costing significant cap but not true free agent dollars, and aged enough to be in their prime and to have had a couple years to fill out a real team around them.
Since 2000, 5 times a team has won Cup within 5 years after losing in the SCF. Only twice during that same period has a team won a Cup within 5 years of having 1st overall pick. So it certainly does seem that making long playoff run and falling short is better path towards winning a Cup than being crappy and getting top pick is.
I do think there is some truth to your point overall, though. Almost every team that won on this list did so on the back of a top 4 pick they drafted. But obviously most of the teams that are drafting in the top 4 have not won the Cup. Drafting in that spot might be the only real way to win in the current NHL now that you can't buy Pronger and Niedermeyer in free agency or hit on a bunch of Europeans and Russians other team's didn't properly scout with later picks. But even then most of the teams picking at the top aren't escaping the bottom of the barrel if their pick disappoints and aren't winning the Cup even if they got a superstar.
Which is why I'm at where I'm at with my Blues fandom. I feel like it's going to be really tough for us to win another Cup without hitting the bottom at some point, but I can't bring myself to actually want us to do that and really rebuild. I think actually winning the Cup has allowed me to be a bit more zen about the situation where I can better enjoy the Blues special, teams that are pretty darn good but not good enough.
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