MacBook Air won’t reinstall Mac OS

https://mrmacintosh.com/how-to-fix-the-recovery-server-could-not-be-contacted-error-high-sierra-recovery-is-still-online-but-broken/

Regardless of the info in the install log, what sort of error message is the Mac presenting to you when you try to reinstall macOS?

Do you know what year the MacBook is?

If not, you can plug in the serial number at Apple Coverage lookup and it will tell you. With that info we can advise the best route to reinstall an OS.

The serial number can be found on the bottom of the computer.

Check Your Service and Support Coverage - Apple Support

If you have access to another Mac, your best option may be to create a bootable installer on USB flash drive and use that to boot the old laptop and install macOS. However, we must know exactly which year that MacBook is so that we can suggest the best macOS version to install. Sierra is known to be problematic when it comes to downloading. You may want to use a newer OS.

How to reinstall macOS - Apple Support

Have you tried using a different recovery boot option?

If you restart with option-command-R you should have the opportunity to reinstall the latest version of the OS that is compatible with the computer.