iMac (5K, 27-inch, 2019) as extended display for MacBook Air

No Apple here, just other users like you.

I think you got some bad info on your call.

You can connect any of these older iMac models to another Mac from 2019 or earlier for use as an external display. Learn how to identify your iMac model.

• 27-inch iMac models introduced in 2009 and 2010: Use a Mini DisplayPort cable to connect your iMac to a Mini DisplayPort or Thunderbolt or Thunderbolt 2 port on the other Mac.
• All iMac models introduced in 2011, 2012, 2013, and mid 2014: Use a Thunderbolt cable to connect your iMac to a Thunderbolt or Thunderbolt 2 port on the other Mac. If the other Mac uses Thunderbolt 3, you can use the Apple Thunderbolt 3 (USB-C) to Thunderbolt 2 Adapter with your cable.

Target display mode requires macOS High Sierra or earlier on the iMac used as the external display. It does not work with macOS Mojave or later.

Your iMac is too new to be used as a display.

Use your iMac as a display with target display mode - Apple Support

If your MBA needs an external display you will need to buy one. Your iMac was never designed to be an external display therefore it will not support Target Display Mode. If you want to look for third party solutions you are welcome to.

You could use Screen Sharing, but it won't work as an extended display, and can be slow.

Share the screen of another Mac - Apple Support