The reason why it should be different until the dying moment is because undeserved grace was pressed upon us, therefore we are always right to be part of the process of pressing it upon others. With care of course, that we don’t get so torn up with refusers of pearls that we can’t minister anywhere else. Jesus had infallible perception of who would never believe, and we do not, and therefore we would be exercising hard hearts by taking that attitude towards all unbelievers.
Also “God doth take no pleasure in the death of the wicked.” The sadness does not need to drag us down forever, however to express it in some measure is not an unscriptural thing (as long as we don’t let negative scriptures completely blot out our perception of positive ones).
Excellent answers, all.
But my point still stand, that this isn’t emphasized in Scripture. We are to preach the gospel, but we aren’t to compromise and/or beg people into the Kingdom.