My MacBook Air starts up to a black screen

Please refer to If your Mac doesn't start up all the way - Apple Support, specifically, If your Mac starts up to a blank screen - Apple Support.

I realize connecting the external display is helpful to see what's going on, but it complicates troubleshooting because merely connecting it causes interaction with the Mac's built-in display. Try to get the built-in display to work normally first by following those steps: power the Mac down completely, power it up again, try to start it in Recovery mode, repair the startup disk, and finally reinstall macOS. Exhaust each one of those steps before arriving at the conclusion:

If your Mac continues to start up to a blank screen, it might need service. Get service.

It sounds like a logic board issue. I would have thought it is the display at first, but when you mentioned that it turns black even on an external display, that indicates some deeper issues.

I would bring it to an Apple Store to have them run diagnostics.

getsupport.apple.com