That would not work. Just because a callback is registered to a hook doesn't mean that the hook will be triggered when expected. The only thing that would sort-of-kind-of work is using the 'shutdown' hook, which you mentioned before:
add_action( 'shutdown', function() {
if ( !did_action( 'template_tag_like_hook' ) )
echo 'Problem.';
} );
Of course, this would be printed at the very bottom, after the </html>
tag, on the front-end (since that's where template tags are normally used), which is not of much use.
You could try to store the message in wp_options and then display it in the admin area, but that would open a whole new can of worms: invalidation, caching plugins etc.