Yes... some of the CofC's do that, but many more do not... at least in my experiences in two different cities in Texas...
Just as some of the Baptist groups do that, and some of the Pentecostals, etc....
I know the history of "the church"... and there definitely was a tendency to think they were the "true" church of Christ... hence the name.
It is the reason quite a few churches have taken the name CofC off their building, because of the negative stereotyping.
I saw one in Lubbock that had on their building "A church of Christ meets here"... I thought that was pretty good.
There are different levels of "strictness" in CofC's... just as there are in any other denomination... There's one in Allen (where we live) that is an "anti" church... no Sunday School classes, no kitchen allowed in the building, no missionary support, dress codes, etc... Then there are some that are pretty liberal about most things, but acapella only, no intruments. Then there's us.... two services on Sunday morning, one instrumental, one acapella. Sometimes people come to worship in shorts and flipflops.
There are three churches in a row on our street... a Presby, a Methodist, and a CofC.... we have, on several occasions, gone together with the other denominations in community events and such.