Thanks for the fast response, ZGani.
Turns out I made a dumb mistake and was updating a previous installation of my site – not the live site. This explains why I wasn’t seeing any changes online.
I agree that 2.3.3 to 2.8.4 is a very big leap, but I’ll be sure to back-up properly before diving in again. Luckily, the site is pretty simple and doesn’t use many plugins, so it should be pretty straight-forward.