Do some MacBook airs not allow external display? Do some only allow built in display?

All notebook models allow the connection of an external display - one way or another. See macOS User Guide. Also, see Use external monitors with your Mac and Get help with video issues on external displays connected to your Mac. Be sure to determine what cable connectors are required for the computer and the external display. You may have to get adapters to use with the cable. See About Apple video adapters and cables and Apple Thunderbolt cables and adapters.

Does your computer have USB-C ports? Recent MBA models only support 60Hz horizontal sync supporting monitors that also can support the resolutions the MBA supports. I don't know which MBA model you have because it isn't stated in your topic post nor in your profile.

When you bought the external monitor did you inquire of the vendor whether the monitor would work with your MBA? Did you bring your MBA to the store to test that the monitor would work?