If I conspire with Jane, and Bob conspires with Susan, then both I and Bob are conspirators; but we’re not co-conspirators. However, Bob and Susan are co-conspirators; they conspired with each other.
Thus “co-conspirators” conveys the clarity that two parties have not merely conspired, but conspired with each other.
I agree with you in principle, as I too enjoy the elegance of the usage, but I do not think the majority of english speakers employing the term are using in in that sense.
To the contrary, I checked the usage in the Corpus of Historical American English (COHA) and this usage is commonplace. People use “co-conspirator” to describe a conspirator who’s involved with either a previously mentioned conspirator or previously mentioned conspiracy. If you’re a “conspirator” you conspire in general, but if you’re a “co-conspirator” you’re part of the conspiracy in question, or more often, conspiring together with the other person or persons being discussed.
Conspirator is a bit like “member of a class” while co-conspirator is like “classmate.” Common usage does support it.