In fact, there were a lot of black actors back then who also should have sued Warner Brothers too for defamation of character. Unfortunately, they didn't have Bing's clout. There's a difference between using a famous face as a caricature and using it as a means to debase someone. In this Family Guy era, we see celebrities and politicians trashed all of the time, and there's not much the victims can do about it (for instance, Carol Burnett tried to stop Family Guy from using her Charwoman character in an R-rated parody, and lost her case. I think she had a point. That character is part of her legacy and she has a right to try to protect it, in my opinion). But back in the early 20th century, standards were different, and it's not surprising Warner Bros. backed off of caricaturing Crosby...for a while.
As to whether or not Crosby himself was a good or bad person, it depends on who is talking about him. A couple of his sons described him as a cold disciplinarian. One of his sons, however, denied that and said that Bing was a great father. And his second family of children (those he fathered with second wife Kathryn Crosby) were devoted to him. There was a story in the newspaper many years ago in which Bing said he hoped to not repeat the mistakes he made with his first family. So clearly there were some issues, issues that Bing came to regret.
To my mind, Bing had some justification to be protective of his image, but Warners had the right to freedom of expression. It's a paradox that plagues show business to this day, and likely will never be fully resolved in any instance in which such a situation arises. As usual, the only clear winners will probably be the lawyers.