I don't see them as a weak cup winner. I just see them as a team that wasn't build to last.
Carolina really capitalized on the 2004-2005 lockout. Lots of marquee names changed places prior to the '06 season, and the Hurricanes substantially upgraded their roster by signing Matt Cullen, Frantisek Kaberle, Cory Stillman, Oleg Tverdovsky, and Ray Whitney. They also made savvy off-season trades for Mike Commodore and Martin Gerber, both of whom were huge contributors. In-Season, they traded for Doug Weight, and later Mark Recchi (after a scary injury to Erik Cole prompted the need for another forward).
The Hurricanes were wretched in 2003-2004, so the modern day equivalent to this scenario would be if the 2020 Ottawa Senators added a bunch of veteran players this off-season and proceeded to win the 2021 Stanley Cup.
While I've seen the 2006 Hurricanes called weak a lot, I'd argue they have one of the absolute best forward groups in recent memory. Keep in mind 2006 was a high scoring season, but not many teams have been able to sport two 55+ point players (Weight and Whitney) on their third line.
Most Common Playoff Line-Up
Stillman - Brind'Amour - Williams
Cullen - Staal - Recchi
Whitney - Weight - Larose
Ladd - Adams - Adams
Hedican - Commodore
Kaberle - Ward
Westley - Wallin
Ward
Gerber
The defense was definitely meh, but overall were solid in their own end. It was full of hard blue collar guys, with Kaberle being the more offensive player in the bunch (Tverdovsky was a healthy scratch by the time the playoffs rolled around).
Overall, my takeaway is that it was a team full of veteran guys you really wanted to see win the cup (Brind'Amour, Weight, Westley, etc.), but the team wasn't built to last outside of that one year.