From my quick research, the last team to win a Stanley Cup that DID NOT have a 'home grown' top-3 overall draft pick on their roster was the 2008 Red Wings. It is up for debate how much each player contributed to their team's success (Seguin). It is also worth noting that in the past 10 years, 3 teams account for 8 of those championships, but that doesn't change the fact that high draft picks are major contributors to the teams that are regularly contending for a Stanley Cup.
2018 - Capitals; Ovechkin (drafted #1)
2017 - Pens; Crosby (#1), Fleury (#1), Malkin (#2)
2016 - Pens; Crosby (#1), Fleury (#1), Malkin (#2)
2015 - Blackhawks; Kane (#1), Toews (#3)
2014 - Kings; Doughty (#2)
2013 - Blackhawks; Kane (#1), Toews (#3)
2012 - Kings; Doughty (#2)
2011 - Bruins; Seguin (#2)
2010 - Blackhawks; Kane (#1), Toews (#3)
2009 - Pens; Crosby (#1), Fleury (#1), Malkin (#2)