They do not even have the support system of a family or the personal status of Yichus ancestry. In time, if they do marry and have children, they will build their own support system. Their children will eventually marry into the body of the Jewish nation. Over many generations, they will gain ancestry. The convert symbolizes the most individual component of our society.
Society is their family.
Society must be their support. If a person steals, it shows a lack of trust and faith in G-d's ability to maintain His world. However, even if the victim has died, because every Jew must have a relative no matter how distant, it is possible to make restitution by returning the stolen money or object to the relative. The convert is the only exception. If he died without every marrying or having children, he has no heirs.