No you cannot boot your MacBook Pro using a connection with your MacBook Air.
What you can do with the connection that you had mentioned is have your MacBook Pro enter the ➔ Target Disk Mode. All this does is transfers data from one Mac to another. In your case it would transfer data from your MacBook Pro to your MacBook Air. Just data transfer. This still would not solve your issue. This is because you have no visual on your MacBook Pro. Its a shame that your external display is not working anymore with your MacBook Pro.
If your external display was still useful then you could have run Safe Mode and even macOS recovery mode to reinstall macOS on your MacBook Pro and address issues that way. Can't do this either due to no visual at this time.
I recommend that you have your Macbook Pro examined at the Apple Store or at an Authorized Service Center when they open back up again. You may even contact ➔ Apple Support
Axel F.