I removed WP SC and got a message from WordPress saying that indeed it was still active and I had to remove code from .htaccess, which I did. I'm not convinced that it's gone though (see below)... Just turned off Cloudflare as well and flushed the cache.
I made the menu changes as per the codex link (thanks for that). Wasn't sure what menu to enable to work on mobile only so I enabled the extra one (nothing happened). Then enabled the top one and of course got the entire 30-category menu above my home page header. Disabled both of them, and... it's still there. I assume that this was cached and now won't go away. Any hints at all appreciated if you get the chance (realise this is well outside your 'remit'...)