The reality is that most americans don't know the details of the case.
The problem is that they think they know the details but are in fact working on incomplete or wrong information. That's not to say they are stupid (they aren't).
The perception is more important than the reality (and in many ways perception is reality). The fact that there is a good reason they polled that way (uninformed) isn't as important as the fact that they polled that way. People are not suddenly going to get informed and change their minds when this ends.
As you say, the majority of americans don't know the facts. What you don't grasp is that it doesn't really matter, from a practical standpoint.