hook_block_view_alter() is not invoked
...but the hook implementation from my theme never gets called:

hook_block_view_alter() will not be called for themes; it is invoked only for modules. This is why your hook is never invoked.
To implement this hook, you need to create a custom module. If you need help to create a module you can check here to start.
I strongly suggest to read block_view_alter docs too.
Look at Theme Hooks to see what you can 'hook' in your theme.
For more information about hooks look here.