Unable to re-install OS after remove all contents by disk utility

You need to erase the whole physical drive instead of just a macOS volume. Within Disk Utility you may need to click "View" and select "Show All Devices" so that the physical drive appears on the left pane of Disk Utility. The physical drive will be identified as something like "Apple SSD....". Erase the physical drive as GUID partition and APFS (top option). After this you should be able to install macOS.

If your MBAir is actually the 2018 model instead of the 2017 model, then you can instead "Restore" the firmware which will also push a new clean copy of macOS 12.x Monterey onto the laptop.

Revive or restore an Intel-based Mac using Apple Configurator - Apple Support

FYI, here is an Apple article with instructions on things you should have done before erasing the Mac:

What to do before you sell, give away, or trade in your Mac - Apple Support

If you have the 2018 model, then the easiest & quickest method is to use the "Restore" firmware option by using your 2021 Mac using the article I linked in my previous post.

The instructions for reinstalling macOS is included in the other linked Apple article I provided for preparing a Mac for resale. The only difference between my instructions and those instructions are I like to erase the whole physical drive on an Intel Mac instead of just erasing the hidden Container.

Here is an Apple article focusing on just reinstalling macOS from Internet Recovery Mode (make sure to follow the link within this article to "erase the disk" or use my slightly modified version from above for the erase step):

How to reinstall macOS - Apple Support

When erasing the drive or hidden Container, you can either keep the original name or use "Macintosh HD" or "Macintosh SSD" where the former is the macOS default for naming the boot volume. After reinstalling macOS you will be greeted by the Setup Assistant asking you to configure the new OS. Just select the language and click next so you are provided access to the Apple menu so you can select "Shutdown". This will allow your brother to go through the setup process himself to customize the OS to his liking.