This is an idea that was mentioned in #1278 and could be part of the #1839 meta-issue...
So the idea is basically to have an additional "Place in layout" action available in the drop-button in the "Operations" column (mockup):

...this would pop up a dialog asking which layout to add the block to, and upon selecting a layout, would redirect to the "Manage blocks" page for the selected layout, perhaps with a "Select region" prompt?
Recent comments
Hi ian, so, in your case all the other admin pages work fine, including the status page, only admin/reports/updates fails? But you can access admin/reports? Weird... I...
Update Report thows "Access denied You are not authorized to access this page."
The File (Field) Paths module should be able to move existing files. I've not tested it, but the module description says: Retroactive updates - rename and/or move...
Moving from /files into subdirectories
Yes indeed. We are exploring a few other more costly options, but as we are a low-resource start-up, we could save a lot of money by integrating Backdrop, CiviCRM and Ubercart for our membership...
UberPOS for Backdrop?