I've never heard Fall Out Boy--I'm a metalhead/proghead so my knowledge of punk is limited to what I listened to in my teenage years, which was mostly mainstream.
What struck me about American Idiot was that they channeled raw anger in a way I hadn't heard from most mainstream punk bands. They're no Sex Pistols, but they took advantage of the anthemic medium in much the same way. It's not just that "Holiday" is catchy, but that it's intensely emotional. My reaction, even just taking into account the melodies, was the polar opposite of yours. In terms of music, I thought it was impeccable. On "Jesus of Suburbia," the short melodies transition seamlessly into each other to paint a diverse, ever-changing soundscape. Boulevard of Broken Dreams really does sound broken, and Whatsername sounds properly nostalgic. "She's a Rebel" is one of my favorites, because it's so simple you'll probably remember it the first time you hear it--like you brought up, it reminds me of protest songs.
Concerning lyrical themes, like I said, I'm a metalhead and proghead so maybe I focus on different things. Lyrics don't always matter, but when bands want them to, they should, and in a concept album (what American Idiot is) they are impossible to ignore. Green Day was trying to tell a story.
Around the same time I was listening to American Idiot, I happened to pick up my first Dream Theater album (they're now one of my favorite bands), Metropolis pt. 2: Scenes from a Memory. Of the two albums, the only one I continue to listen to is SfaM; it's much more well-written and much less juvenile overall. But the Dream Theater album
(here's a sample, if you're interested) primed me for American Idiot. All the words meant something in the context of a larger story, which at the time wasn't something I was familiar with. Subsequently, I wasn't expecting a pop-punk band to write 9-minute epics or delve into political themes, and when they did it reminded me of my newfound love of storytelling in music. Honestly, if you know of another punk band that has done anything like this, please tell me about it, because I would like to hear them. Green Day's no Dream Theater, but the fact that they even attempted something as ambitious is worth mentioning.
Maybe I feel this way because punk's not my genre anymore, and it wasn't when I listened to American Idiot, either. I had grown out of it and felt it had nothing worthwhile to offer. I wouldn't have expected Green Day, of all bands, to throw me a curveball, but it did, and I just respect them for that. When people bash Green Day for being emo and such, they speak a language I don't understand--I just know that it's the first mainstream punk that actually seemed like it had something relevant to say since Bad Religion. And damn, can they tell a story. Concept albums are commercial suicide, and somehow Green Day got people interested in them. To me, that actually brought punk some credibility.
Keep in mind, about the only political issue I agree with Green Day on is their anti-war stance, so it's not exactly like they're pandering to my views, either.