"It is as firm in its conviction that we are "Christians only" (a slogan from the first generation of the movement) as it is in its rejection of "we are the only Christians."
Frome my experience, many of them do believe they are the "only Christians" (and that other denominations have gone astray) - though they are sort of quiet about it.
How many churches were purchased by the blood of Christ?
To be fair, not all of them do (and AFAIK, only some within the Churches of Christ). Of those that do, it's my understanding that they reject as invalid any baptism that was not performed by one of their preachers, and more explicitly any baptism not performed for the expressed purpose of remission of sins. Since baptism for them is absolutely essential for the entrance into salvation, invalid baptism = not saved/regenerated, thus claims of salvation outside of baptism into a "Church of Christ" are generally looked upon with deep suspicion and distrust.
However, in their defense, as the Churches of Christ are not a denomination per se lacking any overarching theology/creed to bind them together, your local CoC congregation may believe differently.
A recent sermon I heard at a church of Christ was about the need to read and study the Bible following the Berean example given in Acts 17:11. The preacher specifically said that we should question what we are told, and study the Bible to find the truth. He said that no one should believe something simply because he said it, or because any person said it, or simply because it is what our parents taught us, or simply because the church teaches it, but because it can be proven by the inspired word of God. (Galatians 1:8)
He mentioned a Gallup poll where a percentage of people in this country actually believe that Billy Graham delivered the Sermon on the Mount, and that Sodom and Gommorah were husband and wife! He said we were being destroyed for lack of knowledge of the word of God (Hosea 4:6), and that the remedy is to read and study the Bible, and to encourage others to do the same. (2 Timothy 2:15,16)