First: You can have a great responsive site loaded with min-widths or max-widths or even mix them up in the same media rule.
People have different approaches when writing a responsive style sheet.
Having your media-queries all written with:
@media screen and (min-width: 320px) {
}
@media screen and (min-width: 680px) {
}
@media screen and (min-width: 1024px) {
}
Means that this CSS will apply if the viewing area is greater than 320, 680 and 1024 respectively.
Altough if you have:
@media screen and (max-width: 320px) {
}
@media screen and (max-width: 680px) {
}
@media screen and (max-width: 1024px) {
}
Means that this CSS will apply if the viewing area is smaller than 320, 680 and 1024.
Second: You can even have a more specific media-querie saying:
@media screen and (min-width: 680px) and (max-width: 1024px) {
}
which states: If the viewing area is between 680px and 1024px those styles will apply.
Wrapping it up, if you don't see many web sites with min-width media-queries it doesn't mean they don't work as good as the max-width ones. They do, maybe the person who wrote the responsive bit was more confortable using max-width.
EXTRA EDIT FOR CHARACTER LIMIT