iOS 10 Proactive App Suggestions are Missing
I've figured out how to re-enable the proactive app suggestions in iOS 10:


Go to Settings → General → Spotlight Search
Turn off the top three options:


Siri Suggestions
Suggestions in Search
Suggestions in Look Up

Go back to the main Settings menu.
Close the Settings app (go to the Home Screen).
Lock the phone (turn off the screen by quickly pressing and releasing the Sleep/Wake button).
Unlock the phone.
Go back to Settings → General → Spotlight Search
Turn on the three options you turned off above.
Lock the phone (turn off the screen).


Now, when I plug in headphones or connect a bluetooth speaker, the proactive app suggestions show up as they did in iOS 9!

I found the basics of this procedure in the comments at this reddit discussion.