Apple Screen Sharing Only Controls One Display?
Alright. The issue is a stupid one which is not entirely evident. When setting up a Mac with this many screens, be careful with the arrangement.  When arranged all in a row, even a 1 pixel overlap will cause the overlapped screen to take over all of the others and cause the behavior I stated above.  It is not evident from the desktop. On a gut feeling I ended up dragging all of the screens around to make sure they were entirely aligned and as far over as possible from the previous screen.  It turned out the first screen was the slightest bit overlapping the second.

I'll leave this question/answer up for anyone else who might encounter this bizarre issue.