As I understand it, lots of prominent Catholics have been through this process, including several Kennedys. The first marriage can be annulled, even when several children have resulted from it, and the second marriage then be declared valid in the eyes of the Church.
To the non-Catholic, such as myself, the entire process looks like a hypocritical end-run around the no-divorce stance of the Church, as it it time-consuming and expensive and therefore not readily available except to the rich and well-connected.
The Church has nothing to gain by granting annullments. Strong families are an important part of the foundation of the Church. Agreeably, there has been abuse of this by politically corrupted clergy, and there have been some flimsy cases to be sure, but it doesn't make the concept an enemy of Christian theology. If the Church decides the marriage didn't exist in the spiritual sense, then it didn't exist. There was a union, but not a marriage.