I"m sure the amount of information made available to us at this point cannot allow for a perfect representation of what Mather would have thought. The following paper is my best effort at answering what the purposes of the Puritan Movement, the challenges it faced and the eventual final opinion of Mather on the Puritan Movement. .
The Puritan Movement was based on many ideals and beliefs, not to mention the fact that it was quite literally a movement of people to America from Europe. The Movement emerged from The Church of England and was greatly led by John Calvin and his strong beliefs. Seeking refuge and religious freedom to believe what they want without persecution, the Puritans set up a whole system of ideals in the new America. One of the largest beliefs of this culture was the idea of predestination. They thought that people were born into "original sin", meaning from birth people were already damned with sins. The whole purpose of that person's life from birth would then be to lead the perfect life and possibly earn the right to be redeemed ("saved"). The Puritan Movement which publicly criticized the Catholic Church tried to build a culture void of corrupt sinful actions such as drinking, gambling, swearing and so on. It was a culture where religion was the entire meaning of life and those who did not believe this were ostracized. In this kind of community if religion is this important, the people that are higher up in the church are going to have more power naturally. These people were honored and respected, and over time the priests, pastors and so on became quite powerful. The Puritan church was set up so that in order to fill the ranks of a pew in the church, a person had to first be considered "noticeably godly" or be part of the "elect". These were people that were believed to be "saved" and led perfect lives completely void of sin. So the only people allowed into church were the "elect" and the religious leaders, which would eventually lead to being a member of church giving you an authoritarian position.