Xbox – No audio in Overwatch for Xbox One

You are correct. Setting your settings in regard to what others can see to "Block" will prevent them to see the type of content you blocked.

The reason why you are still seeing it is because the functionality to gather all your activities cannot be turned off. Hence your feed is still assembled but not shared (or partially depending on your settings). This will enable you to share past activities should you wish to toggle the settings again.

So I kept putting off the whole uninstalling the game and resetting the console because it was a pain and played something else. However, the problem was fixed when I went back to play the game at a later date and there was an update. I'm guessing this refreshed the install somehow in the same way that reinstalling the game would and that there had also been a console update which had reset any cached information.

If anyone does try the "official" method listed above and finds it to work (rather than waiting for updates), then write it here as an answer and I'll change the accepted answer to be that one.