Spotlight search on iOS not showing options to search web/maps/App Store unless it has results from other apps
Upon a little testing around, I have discovered the solution. Go to Settings > General > Spotlight Search and disable Bing Web Results. There must be some problem either on Microsoft's end or in the way the iOS software gets it, and this interferes with Spotlight from doing what it needs to do.

Hopefully Apple will fix this and it can be re-enabled with iOS 10 or a future iOS 9.x update, or it will be fixed on Microsoft's end, if the issue becomes known to them.

It's unclear to me if this affects everyone or just a small number of users. If you're not sure if you have this problem, test it out as per the original question. And please comment if this does or does not fix it for you.