Ok, I checked an older computer that isn't getting updates, and my first post was correct. Software Center had an option 'Automatically check for updates' which was an on/off toggle. If you selected 'off' solus-sc was simply off. It was not running, and therefore also not checking for updates until the user started SC, which was also obvious since it even said 'checking for updates' on the bottom of its window, which took a bit, instead of having them instantly listed.
DiscoverNotifier is always on, I guess. I can see it in the process list pulling between 10 and 300 MB of RAM when I haven't even touched it. This despite me telling it to notify me 'never'. It seems there is no option to change this. I also notice its periods of activity near network activity, so I guess it just checks for updates whenever it wants, like Windows. That's how I got a random notification of updates a good while after boot.
As much as I would like Discover to gain an 'off' switch, I don't expect anyone to add that. It's been around a long time and it wasn't made for Solus. I guess I didn't realise what 'made for Solus' meant. Whoever wrote Software Center (Ikey?) had more respect for their users. Ah, well. I understand that no one on the Solus team had time to write a new SC, and I respect that. Now I hope the upcoming Nemo>Dolphin switch doesn't have any unpleasant surprises.