There are some observations I would make.
Most members of churches of Christ would not know what Cane Ridge was, much less consider it the beginning of the church in America.
Cane Ridge was a multi-day, multi-denominational event with preachers from a variety of groups, including Presbyterians, Methodists, and Baptists.
There is no reason to deny that the Spirit was at work in that place. Whether everything that occurred was a working of the Spirit or not is questionable.
If churches of Christ claim a date for the beginning of their movement, it would most likely be the 1832 meeting between Campbell and Stone where they decided to join their two groups which had begun independently. Some might point to the rejection of Campbell's followers from the Redstone Baptist Association (1824), or perhaps even the dissolution of the Mahoning Association as the "beginning" of a separate group.
These two groups (Stone's and Campbell's) were not the only groups to come to similar understandings of what the church should be, but they evidently provided the impetus to the development of what is called the American Restoration Movement, or Stone-Campbell Movement.
The church of Christ penchant for more cognitive approaches to Christianity is likely an artifact of Campbell's style, but since most members of churches of Christ couldn't tell you what Campbell thought nor his style, these are not determinative of current belief. More likely, the statements of Jesus that essentially point to his words and deeds as reasons to believe, and Paul's rational argumentation style provide the current basis of a cognitive approach.
However, to acknowledge that churches of Christ take a cognitive approach to Scripture does not mean that they deny the operation of the Spirit. There are some, no doubt, who do. There are many who do not. Churches of Christ do not fit neatly in a box.
Churches of Christ are not Calvinist, and to that degree, they deny that the direct operation of the Spirit is needed for conversion. That however, is not the same thing as denying that the Spirit operates in the "real world."