Given that Edward has indeed turned out to be a vampire, the legend Jacob told is probably true. So even though Jacob doesn’t put any stock in the legend, Billy sees Bella and Edward’s relationship as something terrible and dangerous. Though this poses problems for Bella and Edward, it also illustrates how much Billy cares for Bella. He might be a pretty distant presence in her life, but he still doesn’t want her getting into danger—especially if she doesn’t understand the danger. It also means, though, that Billy judges Edward based on what he thinks Edward is, rather than on how Edward acts.