He had 2 living sons he didn't care about

Unfortunately, that's very common with parents who lost a child. The fact that the deceased kid exists only in their memory and imagination sets an impossibly high idealized standard that the surviving children can never live up to.

I don't see this as being a major theme in Jacob's Ladder apart from how it relates to the broader theme of accepting death and Jacob's own inner turmoil, but there have been some good films made that focused on neglected (and in some cases despised) surviving children, such as Ordinary People.