My Fitbit inspire display only lit up when I pushed the side button. So I decided to try a restart. While I was holding the button down this little light bulb with a line through it showed up. I tapped the light bulb and the line disappeared and the bulb lit up. My inspire now lights with movement like before. I never knew about this light bulb, and do not know how it got turned off.
If you notice, right under the light bulb is the label 'Screen Wake'. Many people don't realize that setting is there, and it does seem to get turned off accidentally - maybe when going to other settings such as doing restart as how you found it.