I did some experimenting, and I found out that the installation method I mentioned in the original question works with the 8.0.10. Or at least - it partially works - the application only worked after I executed
chmod -R 777 /usr/local/zy-pkgs/gui/ownCloud
as a last step. Without this I was getting a blank page.
Of course I highly discourage doing this as this allows anyone to read, run or edit any file in the directory, which is a serious security breach.
Currently I'm experimenting with the ownCloud built-in upgrade tool, to see how much will it allow me to upgrade the application, then I'll try installing the highest working version manually, and I'll try finding the file that needs some special attention when it comes to permissions.
Update: The ownCloud successfully updated to 8.0.13. Now the updater suggests upgrading to 8.1.8. Trying it now.
Update 2: Version 8.1.8 doesn't work, trying to open it in the browser fails - a blank page is shown. Chmod'ing the ownCloud directorry to 777 doesn't change anything. The 8.0.13 is probably the newest version that can work on these devices, and for now I'll stick with it.
Update 3: Version 8.0.13 can't be installed using this method (blank page despite chmod'ing whole directory to 777). The only possible way to get 8.0.13 up and running seems to be installing 8.0.10 and then upgrading using the built-in tool.
Update 4: I managed to get version 8.0.13 work from clean install after chown'ing the ownCloud directory to root:root and chmod'ing it to og+rw (I know it's probably not an optimal solution from security standpoint, but that's still way better than 777). Then I updated it succesfully to 8.0.13 using the built-in updater. The whole process of installing the new version on the device is as follows:
(Connect to the NSA as root using telnet - you first need to enable
in the administrator's webpanel. Then you can run the commands.)
cd /usr/local/zy-pkgs/gui
wget https://download.owncloud.org/community/owncloud-8.0.13.tar.bz2
tar xjvf owncloud-8.0.13.tar.bz2
cp ownCloud/config/config.php owncloud/config/config.php
vi owncloud/config/config.php
(To set 'version' to the correct value for the downloaded one - 8.0.13 in this case)
mv ownCloud oldCloud
(Just to be safe, keeping the old original version at hand for now.)
mv owncloud ownCloud
(Putting the new version in place of the old one.)
chown -R root:root ownCloud
(Changing ownership to root:root, the way original ZyXEL apps are)
chmod -R uog+rw ownCloud
(Giving permission to read and write the files to the system users,
this is needed, because otherwise the app won't work - an empty
page will be shown.)
Next step is opening the app in browser, by visiting
http://nsa's_ip/pkg/ownCloud and following the instructions
displayed on screen. Feel free to remove the oldCloud directory if
everything worked fine. To do this run the command:
rm -rf oldCloud
DO NOT update to 8.1.X or newer, it will not work on these devices - blank page will be displayed.
Update 5: I have tested the current version (8.0.14) - both the update from 8.0.13 and the clean install process based on the one shown above work. The application is running cleanly and reliably. Still no luck on 8.1.x though.