I don't really think it's old and slow. I think they're soft and mushy in the middle. Bresee has turned out to be way more of a pass rush specialist on the interior than I expected, but it's not just that.
Think about the way the Saints' defense has been trending pretty much since DA returned as the DC. Even back in the Kenny Vaccaro days they were basing out of this nickel big/4-2-5 look and putting a premium on pass rushers up front. The returns were great--the Saints were fast and athletic and able to match up in a league full of 5,000 yard passes and rules engineered to favor explosive passing games. This also helped them in the draft, where instead of needing to find a premium, 3-down SAM LB, they could draft a tweener like CGJ in the mid rounds and have a 3-down player at nickel.
But there's been a shift in recent years. Partially due to the type of QBs coming out of college and the league embracing RPOs and a variety of option football. But also just as an overcorrection to the emphasis of defenses as of late toward athleticism over size. The fullback was disappearing, tight ends were becoming more F-type players, flexing out and serving as matchup/size advantages in the passing game rather than in-line blockers in the run game.
Now fullbacks are back, and so are in-line tight ends. Running games are dominating the league and passing is down. The Saint defense, which spent a decade building for this new pass-happy league, is caught a little with its pants down. I think it's this one-dimensionality of the team--along with a little bit of aging and some injuries (as well as some key departures at key positions)--that is taking a toll on the defense.