I know there are some conservative churches of christ, but they all come from that same reconstruction stuff from the 1800s. The disciples of christ come from the same time, also, if my memory serves correctly.
Churches that follow God's Annointed One (Messiah/Christ) will be true to scripture. They will have a strong regard for Christians of past eras. They will teach the true good news that "Christ died for our sins according to the scriptures."
Those called out ones of God's Messiah will not be defeated and need "reconstructing." "The gates of hell shall not prevail against them."
Your memory does not serve you correctly in the history of the church. The church has always been in existence, began in Acts 2. None of the restoration leaders would ever deny that the church has been in existence. Their goal was to speak where the bible speaks and be silent where the bible was silent. The history of the United Church of Christ is completely different.
While you call it reconstruction as a way to discredit the movement, it still embodies the thought or philosophy that the church should be established or constructed on bible principles and teachings. The Church of Christ does teach the good news, the gospel that Christ died for sins and we teach others.