I have resolved this problem so I thought I would post the resolution.
It wasn't a disk issue, it was a folder permissions issue. At some point, all my Shared Folders' permissions got totally messed up. Not sure of the original cause. Possibly my attempts to fix things made it worse, permissions are really complicated.
I fixed it by removing each drive from the drivepool, copying any remaining data off the drive, reformatting, then adding the clean drive to a new pool. Then I created new Shared Folders in the new pool and gradually copied all my data into them, checking permissions as I went. I suspect this is a lesson for any new Windows Server installation: create new shared folders and then copy data into them. Don't try and point straight at your old shared folders.
The fix took ages because I had more data than space on any one drive, so I had to keep shifting data around. Permissions were so messed up there was some data I simply couldn't move off a drive, luckily nothing important. I suspect there could have been file system level problems, because I wasn't able to reformat the drives until I deleted the old partition using disk management.
Since the fix I have moved my Client Computer Backups shared folder back to the drivepool and it seems to be working fine. So I don't think there is any inherent problem with having that folder in a drivepool.