I love ADTR and I love FLL but there are distinct breakdowns on 2nd sucks, It's Complicated, Sticks & Bricks and You Be Tails. Not as many as Homesick or previous material but they are definitely there. And for the record I think ADTR does breakdowns better than most other bands.
You can hardly call the bridges in It's Complicated or You Be Tails "breakdowns," man haha. At least not even close to the traditional sense in metalcore, or what he's referring to I'm sure.