It is necessary to be aware of your spiritual poverty before you can become a Christian because it is the only way that you will appreciate Christ's sacrifice and the gift of salvation. I don't think it's possible for you to accept the gift of salvation without first realizing how far you are from the Father.
The kind of mourning that is necessary for a person to become a Christian is that over sin and separation from God, and the ultimate punishment that comes as a result thereof.
Many Christians today mourn as a result of the sin that they see in the world around them, and also as a result of the trials and tribulations that they endure by merely being children of God. We see in Isaiah that God sees their ashes, mourning and despair but He promises them beauty, joy and praise!