In short, please read Target Display Mode, specifically where it says:
iMac models introduced in 2011, 2012, 2013, and mid 2014
To use any of these iMac models as an external display:
- The iMac used as a display must have macOS High Sierra or earlier installed.
- The other Mac that you're connecting it to must have been introduced in 2019 or earlier and have macOS Catalina or earlier installed.
- The cable connecting the two Mac computers must be a Thunderbolt or Thunderbolt 2 cable.
This simply means your new MBA is too new. I strongly recommend that if you want/need an external display for your MBA to buy one, this is much easier to setup, much more reliable and does not require expensive cabling.