I just did a quick test on my MacBook:
1) Apple menu -> Restart
2) The Restart dialog box appears on the screen
3) Placed mouse cursor over the Restart button
4) Turned off the display by tapping the reduce screen brightness key
5) Clicked on the Restart button
6) MacBook reboots, display turns on, Apple logo appears, login window appears.
Bottom line is that your MacBook Air should always show the Apple logo and the login window after a cold boot or restart even if the Airs display was "turned off" by the OS.
In a dark room look for the backlight to come on before the Apple logo appears. If you don't see that then it is most likely a dead display.
Edit: The above should be undertaken with no external display or adapter connected to your Air.
1800X | 32GB | 1TB M.2 | Vega 64 + 38UC99-W | RX 480 + U2711
1950X | 64GB | 500 GB M.2 | 1 TB M.2 | 1 TB M.2 | GTX 1080 + XB271HU | GTX 1060 + U2713H