The short version of Ottawa's playoff history between 1999-2008 is that they rarely had a #1 center or a #1 goalie and occasionally neither.
2006 felt like an incredible season, it was destiny. Then Hasek got hurt at the Olympics and Chara busted his hand on Eric Cairns head. They faced Tampa in the first round and rolled over them because Tampa was clearly beaten down by a season of atrocious goaltending. Had the season been 86 games long, Tampa would have missed the playoffs. Ray Emery was awful in the Buffalo series (0.864 SV% and allowed 3 OT goals on 4 OT shots). Buffalo diced him up (slow lateral movement) and only barely won the series in a bunch of 1 goal games. This Sens team with Hasek was probably Ottawa's best chance at a Cup.
2007
The first 1/3 of the season is a credit to Ray Emery. The team was playing terribly and tended to start slow every game. He probably had something to prove after his terrible postseason and the signing of Gerber to be starter. Gerber's terrible string of games at the beginning of the season opened the door for Emery. In the 1/3, Emery was stellar in the 1st period and solid after that. For the rest of the season and playoffs, he was just really solid; stopped the shots he needed to, bounced back after losses, etc. The team starting firing on all cylinders after Christmas through the end of the year. They blew through the first 3 rounds with Spezza & Alfredsson gaining the zone fairly easily and Fisher, Neil & Schubert pounding opposing d-men on the forecheck across the other 3 lines. IMO, you could noticeably see opposing d-men developing the yips over the course of the series and that just made things easier for the top line. They were thriving with the NHL's crackdown on obstruction. Anaheim was the first time they'd faced any obstruction in playoffs and they were surprised by it. The obstruction shut down Ottawa's forecheck and Spezza's zone entries. The only two players who thrived were Alfredsson and Fisher, because they equally adept at carrying the puck into the zone or retrieving dump ins.
About the 1 line team comments... look up Fisher, Comrie and Schaefer in terms of regular seasons points and PPG. Then compare them to the 4-5-6 (forward) scorers of a team you think had better secondary scoring. The Ottawa trio probably did better in points, PPG or both. They were quite good back then.
2008
Gerber was embarrassed about losing his job the year prior and Emery had off season wrist surgery. Gerber came out hot, Emery joined a month into the season and Ottawa had amazing goaltending for the first 1/4. Then doubts started to creep in about who was the starter, neither started performing and Paddock was probably making it worse. For the next 3 1/2 years until Anderson was acquired, Ottawa had arguably the worst goaltending in the league. The goaltending let in a lot of deflating and back-breaking goals all season long. Just like Tampa in 2006, if the season had been 86 games long, Ottawa would have missed the playoffs. The went to the playoffs on a death spiral and got swept, which was easy to predict.