Thanks, Estel, that's a wonderful guide.

FWIW, I have figured out what my problem with BackupMenu was in the end. As it turns out, deleting empty /boot had nothing to do with it. This is what the problem was:

I have taken to developing on the device. The package builder uses tar and find with options not supported by our busybox version of those utilities. So I installed the GNU versions (tar-gnu and findutils-gnu). And here comes the foolish part. Instead of adding /usr/bin/gnu to $PATH, I manually redirected tar and find from busybox to the GNU equivalents. Redirecting them back did not help, presumably because I was forgetting something. In the end, I uninstalled everything related to backupmenu and *-gnu again, reinstalled bysybox-symlinks-*, reinstalled backupmenu and voilá, I am back in business!

(And BTW, yes, I ran fsck from my repaired BackupMenu. No problems found )