A third generation iPad Air is capable of update to the most recent version of iPadOS - at time of writing, this being iPadOS 17.5.1. By contrast, an iPad Air1 is limited by hardware to iOS 12.5.7, whereas an iPad Air2 can be updated to iPadOS 15.8.2.
As such, if an App requires iOS 9 or later, any model of iPad Air should be capable of running the compatible App. Perhaps your issue is not related to the installed version of iOS/iPadOS at all - but is instead some other problem.
Consider deleting the offending App from your iPad - and then perform a forced restart before reinstalling the App from the App Store:
Remove or delete apps from iPad - Apple Support
Force restart iPad - Apple Support
If the problem persists after restarting and reinstalling the App, assuming that other installed Apps are working as expected, you might be best advised to seek assistance from the App Developer:
How to contact an app developer - Apple Support
Should you continue to experience difficulties, you may find that instead of using a dedicated App to access your NYT subscription, you may be able to access the same material from the NYT website - using a web browser.