Because all objects have been assigned to all drives already at that point and DP is not re-assigning objects at any time. 

So the solution is to have the 2 large targets assigned to each device rather than load balanced?

Let's say that is a workaround indeed. I believe automating this and making it work for any type of integration or backup would be a pretty large design change. But any idea is welcome on Idea Exchange.

It appears that once a backup specification is created as a load balanced backup, you cannot selectively change any of the individual targets to use a specific device.  So I would need to create a new non-load balanced backup specification and individually assign all targets to specific devices in the attempt to manually load balance the backup.  Certainly not ideal !  Any thoughts?  Certainly I am not the only user faced with this?

I'm afraid that is correct. The other way around it is possible and you will see the following warning.

Which confirms again your statement. Going to load balanced is easier if you think about the datalist format. There it means: just move all devices to the top of the datalist. Moving away from load balanced means that each device that is included in the datalist needs to be moved to a spot in the datalist right before the object you want to assign it too. So slightly more complicated, but definitely possible in my humble opinion. The note I want to add here is that manually editing the datalist is not supported. 

I do not use load balancing a lot, but does DP assign all objects to devices before staring the backup, or does it assign at most "n" objects to each device before starting? I think you can enter a limit for concurrent objects being sent to a device (in drive's "Advanced Options->Settings->Concurrency"), and if so I guess the decision for the rest of the objects would be delayed then. Is that right?

The concurrency defines the number of active objects assigned to the device at any time. So basically the number of disk agents connected. The assignment is happening dynamically. Whenever an object has finished and so a position becomes available, that device can connect to a new object on the list.