I found that in order for the button to appear again, you just need to open VK in a new tab.
So no need to delete cache or cookies, or to refresh the whole profile. Thus I rolled back my old profile, open VK in a new tab, and the button is back again.

Refreshing FF helped, the button is back again, thank you.

And please remove "Fixed" label — the problem isn't solved.

The button disappeared again (and all the other that happened before). I uninstalled the extension, cleared cache and cookies, disabled all other extensions, installed SFN again, but that didn't solve the problem.

Strange, I didn't touch anything, but now everything is back to normal.