Here is the Canes' TOI distribution during the 2005-2006 regular season from the NHL.com stats engine: even strength, power play, penalty kill, overall, and also points.
While Eric Staal led the team in points, Rod Brind'Amour led it in time on ice. He was the top-line center, playing with (I think) Justin Williams and Cory Stillman. Brind'Amour was the shutdown center as well as a secondary scorer. Staal carried a lot of the scoring load as the second center in terms of TOI. I think he played with Erik Cole and Ray Whitney or Matt Cullen. Looking at the "Pos" column, it's clear that the Hurricanes had plenty of options down the middle. To this day Brind'Amour, Cullen, and Staal could be 50 or 60 point centers in the league, and Staal 80 points.
On the power play, the Hurricanes looked to four players first and foremost--Ray Whitney, Staal, Brind'Amour, and Stillman, and after that really spread it around, as eighteen Canes were receiving more than a minute of PPTOI/gm (again note though that that season saw many, many power plays).
On the PK, the Hurricanes trusted all their defensemen, essentially. Seven different defensemen averaged over a minute on the penalty kill per game, the most of which belonged to Aaron Ward at just over four minutes per game. Considering how many power plays there were per game that season, Tom Poti playing 5:30 a game shorthanded as he did this season is clearly not the answer, especially for a guy who's
In terms of overall TOI, Bret Hedican was the only defenseman to break 20 minutes a game, and Brind'Amour and Williams were the only forwards to break 20 minutes a game as well. In the playoffs, only Aaron Ward joined them, as A. Ward-Hedican looks like it became Carolina's shutdown pairing. Mike Commodore really ran up the ESTOI/gm ladder, getting into the "Hedican/Ward" group at 16 minutes or more. Glen Wesley started taking over four minutes of SHTOI/gm as the Canes reduced themselves to essentially trusting 5 forwards and 5 defensemen on the PK. On the power play, they basically played eight forwards and three defensemen (though Erik Cole missed almost the entire playoffs, so make that seven forwards). That doesn't look too dissimilar from what the Caps run: 4F/1D and 3F/2D.
This is how we could set it up:
Backstrom ~ Brind'Amour
Poti-Schultz? ~ Hedican-Ward
? ~ Staal
And that's the main part. The rest is depth. But again, we see no good second line center comparable. The Caps may have better wingers and the best defenseman of the two teams, but they're missing that minute-munching scoring-line center that Carolina had.
And the luck.