I think most of the conservative Republicans I talked with during that time felt that it was fairly clear that Lee Harvey Oswald was involved. We also believed that there was something being hidden. However, the something was probably a second gunman which would have pointed to a wider conspiracy.
Let's look at the possibilities:
1) Lee Harvey Oswald wasn't involved
This clearly opens the door widely to a "name your favorite scoundrel who you believe was guilty" feeding frenzy.
2) Lee Harvey Oswald was involved with two further alternactive:
2A) Lee Harvey Oswald acted alone.
The government seemed to want to come to this conclusion so that it could be asserted that he acted "on his own" with no conspiracy. This was the conclusion of the Warren Commission. However, he could have still acted as an agent for someone else. Cuba certainly comes to mind since Oswald had actively been involved in Cuban politics.
2B) Lee Harvey Oswald did not act alone.
This assumes that there was at least one other gunman. (How many more doesn't really seem to be relevant.)
Of course, this means that there was a conspiracy involving Lee Harvey Oswald. Now we are to the crux of my problem with this article: How would Johnson get Oswald to participate in a conspiracy to help Johnson?
Instead, I would propose that there were probably other conspirators and that the U.S. did NOT want to implicate any foreign power for a variety of reasons.