Alright I'mma start with this Empire! Empire! band and The World Is a Beautiful Place because I feel like I've seen the most about them.
Empire! Empire!, Joie De Vivre, and Everyone Everywhere are probably closest to that Mineral/American Football/Texas Is The Reason sound. All three sound like they're right out of the 90's.
A lot of the bands mentioned in this thread though are more similar to Cap'n Jazz/Braid and some of the faster and more "punk" emo bands than the quiet/sad/subdued stuff.