I've only seen it done in movies about burials in England, that I recall. It could be, however, that it's also part of the custom in Catholic or Anglican burials here, at least for the priest to do; I don't believe I've ever been to one.
I haven't been to that many American burials, and none in recent years. As I recall, though, the coffin is just left at the graveside, or suspended on straps over the grave, to be buried by the cemetery staff after the family and guests have left. However, if anyone wanted to stay and see the burial completed, or toss dirt or flowers on, I imagine you could easily arrange it.
Partly, modern American coffins tend to be metal-lined and extremely heavy, and are often no longer lifted or carried at all, but rolled into and out of the church or funeral home on wheeled stretchers by the funeral-home staff. The pallbearers are often strictly an honorary position, in part out of practicality; at the age at which most people die now, their remaining friends are often too elderly to lift anything heavy. To me that's sort of sad -- I would rather have a lightweight wood coffin and have it carried by people who had known me, and not have an ugly metal stretcher at all. But that's probably wishful thinking.