You're thinking too big.
Some effects are very simple.
Like, cutting out an anime person and putting it infront of a cloud arrangement. Effects can be sentimental, too.
I said Fuel at first, because I was listening to their "Something Like Human" album. And although I had had this idea for a while, Fuel and Madonna seemed to have a nice mix of sentimental and rock-y songs.
When I say "Shove as many effects as humanly possible into an AMV", that doesn't mean I am thinking of eye-bleeding, seizure inducing, or "wow, that's energetic" effects.
There's a wide range of special effects out there, and just as there are different categories of AMVs, different special effects can be used for different "Moods".
Nightowl's "Mad world
" is a good example of sentimental effects.