Why do we keep the anachronism of the royal family? What is their purpose?

The Queen provides continuity. No matter how many political parties are in power she is still there and is a repository of all that has gone before. You need to check your resources again. The royal family is not supported by the tax payers. The Queen is one of the richest women in the world in her own right. What is given to the royal family for their public duties is actually paid back to the treasury by the Queen. She and most of the royal family pay a tremendous amount of money to support their duties.