Church of Christ? That is the church of my youth. I imagine Thompson has probably given it up the way I did, judging from the way his spokesperson is carefully saying that he “was baptised into the church of Christ.” If he were still active, he would have been described as “a member of the church of Christ.”
Members of the churches of Christ have a bad habit of telling all other Christians (including other Protestants) that they are in danger of going to Hell because they are not practicing Christianity in the “right” way. As you might imagine, that engenders resentment among members of other churches. Don’t know if that factors into Dobson’s thinking, but it may be instructive for those who imagine conservative Protestants to be some sort of monolithic group.