Since my original post, BS has increased from 11.0.1 to 11.1. Image Capture is 8.0 (1106).
chuck_3rd: Per you suggestion I began with a Restart. Then I started Image Capture and navigated to a folder containing one raw file (.NEF) that I just took, and clicked on that file to select it. Then I hit the Actions button and ensured that "Delete after import" was checked. Then I hit the Download button. The file was downloaded, the little green check appeared on the Mac screen next to the thumbnail image, but the file was not deleted.
I then tried the Download All button, another copy of the file was downloaded, but again the file was not deleted from the window nor from the camera. Next I unchecked "Delete after import" and tried both Download and Download All: the file was not deleted from the window not camera, but of course it shouldn't have been.
I quit and restarted Image Capture: it displayed the same file without the little green check.
To check the "deletability" of the file, I selected it and used the delete command in IC's Edit menu. This deleted the file from both the window and the camera. So it looks like the only problem is the failure to delete after importing a file, when *** is checked/selected.
Would you like me to try anything else?