I think I was getting notifications for iMessages on the broken restore - not sure now. I didn't notice not getting them.
Unrelated to you:
I had noticed (on my broken restore) that the "myfitnesspal" app was updating with steps that my iPhone is recording, so it seems it might actually be able to access the information, and the steps information is being recorded? It's just the health app can't read/display it. I think the only way to check if steps are being recorded, is make a backup to iTunes, use a backup explorer to see the filesize of the health database. Then make some more steps with the iPhone, do another backup, check health database filesize again... I may try that at the weekend.
However, the 7-minute workout app has an option to enable health data, but it will not switch on.
Does anyone have an app that can read multiple items of health data, to see if it is able to see the data and see if it has the new data since the broken restore?