The song can be easily interpreted if viewed through the lens of social stratification.
The singer's family is fairly well off by Mississippi Delta standards. They're not sharecroppers. They own the land they farm. That the brother and his new wife were able to buy a store in Tupelo indicates that one or the other of them had money or good credit. Papa may have had enough saved up that, when he died, Brother got a decent inheritance. It doesn't sound like Mama and the singer moved from the family farm, but they may have sold or leased some acreage.
Billie Joe, on the other hand, is from Choctaw Ridge, the song's version of "the wrong side of the tracks". Mama says, "Seems like nothin' ever comes to no good up on Choctaw Ridge," and that's not far from "Seems like nothin' no good ever comes from Choctaw Ridge," and if Billy Joe is from there, then he's no good. His family doesn't own their land. They are sharecroppers, if not something worse.
That's why the family is so cavalier about his death. The singer and her brother included Billie Joe in their social circle, but it was a grudging and condescending inclusion.
But, despite their differences in class and status, Billie Joe was in love with the singer and attempted to propose marriage and give her an engagement ring. She rejected him outright and was so enraged at the idea that he would be so presumptuous as to think she would marry him that she attempted to throw the ring off the bridge.
When the preacher saw the couple on the bridge, they weren't jointly throwing something from it. The girl was trying to throw the ring and Billie Joe was trying to stop her. Their struggle is what drew the preacher's attention.
Billie Joe committed suicide not only because of his rejected marriage proposal, but because he realized why it was rejected. He had thought himself in some way equal to the members of the singer's family and worthy of marrying into it. But he was obviously deluded. The singer's family considered him an inferior and always would
The singer's propensity for picking flowers and throwing them from the bridge is her way of atoning for her actions and attitudes. Her rejection of Billie Joe's marriage proposal may have been the proximate cause of his suicide, but the low regard in which Billy Joe and the other folks from Choctaw Ridge were held by her family was the underlying cause. That's why she specifically picks flowers from Choctaw Ridge. Picking them elsewhere wouldn't work.