Not really.
2006 - Canes win with a rookie backup
2008 - Osgood had a career .905 save percentage in the lowest scoring era in modern history
2010 - Hawks win with probably the worst starter in the NHL
2016 - Pens win with the local vagrant who lives at the gas station
2017 - Pens repeat with the local vagrant who lives at the gas station
2019 - Blues win with a rookie who has since lost the starting job
As for the rest of the winners since the lockout, Fleury won it and he's famous for his playoff meltdowns, Giguere and Crawford are very good but also played on comfortably the best teams, Quick and Hotlby have had completely average careers, and Vasilevskiy is arguably the best goalie but also the Lightning could win the Cup with a trained seal in net.
The only team since that lockout that's won the Cup that wasn't also a top 3-5 team in the league outside of goaltending, that won because of goaltending is the 2011 Bruins.
So since the lockout, you've had one goaltender carry a team to a Cup and SEVEN outright bad goaltenders win the Cup.