Problem has been diagnosed: all my sequences and nested sequences do not have the audio panned left or right like they need to, so anything in the left or right channels is heard in both. The solution is to pan everything on the left and right tracks left or right.
The problem now is that this needs to be done not just to the top level sequence, but to every clip inside every nested sequence. FCP7 appears to only be able to change this one sequence at a time, and so I need to systematically go through and do these things in a labyrinth of nested sequences without missing anything or losing track of where I am. Not all sequences are even listed in the list, so I truly would need to go from left to right in the timeline, going deeper and deeper each time there’s a nested sequence, and work my way back up. It would take quite a long time, so I am still searching for a quicker solution. It makes no difference how I export anything, because fixing the issue at the top level doesn’t fix it at the deeper levels and so the problem comes from the bottom up.