MacBook Air software update stuck in a loop "Checking for updates"

You don't state what chip/processor is in your MacBook Air. First check if your Mac is compatible with macOS 27 Golden Gate. That will depend on the specific model. Apple’s current compatibility list is here:

macOS 27 Golden Gate is compatible with these computers - Apple Support

If you are not offered an update in System Settings and/or are unable to install an update downloaded via Terminal, then is likely that you are in Low data mode. Users have reported difficulty upgrading macOS over Wi-Fi in Low data mode, and that toggling that setting OFF enables them to perform a Software Update. 

Turn off Low data mode

• Open System Settings 
• Click on Wi-Fi
• Click on the Details button on the right side of your connected network
• Toggle the setting off

More info from Apple: Wi-Fi settings on Mac - Apple Support

If macOS 27.2 does not appear in Software Update, try the following steps in order.

1. If you are using a VPN, shut it off.
2. Restart your Mac.
3. Go to System Settings > General > Software Update and check again. Press Command-R to refresh that settings pane, just in case.

If macOS 27 still does not appear, you can check which software updates are currently being offered to your Mac.

Open Terminal (find the app with Spotlight or in Applications > Utilities) and paste the following, then press return:

softwareupdate --list-full-installers

If macOS 27.2 appears in that list, you can download the full installer with:

softwareupdate --fetch-full-installer --full-installer-version 27.2

It has been reported that if the update attempts to install but fails before completing, and your Mac stays on the previous version of macOS, that if you are in Lockdown Mode than disabling that will allow the update to install normally.

JustAskinReally wrote:

The install worked (system said something like it was successful) but I'll try it again probably tomorrow and will keep my eyes peeled for the % installed.

To confirm, are you stating that some time after entering the Terminal command (softwareupdate --fetch-full-installer --full-installer-version 27.2), Terminal reported that the installation was successful? That just means you successfully downloaded the installer.

Note that the command is to FETCH the full installer, and in that context, what Terminal calls 'installation' means installing the macOS installer application on your Mac, but not actually running that application to upgrade your Mac to 27.2. To update your Mac to Golden Gate, you need to go to your Applications folder and double click that app (it should be called "Install macOS Golden Gate").

JustAskinReally wrote:

This is to John Galt who suggested I try coming up in Safe Mode (sorry, I can't find the actual response anywhere. I tried it hours ago and got a screen that gives me 4 choices - 1) Restore from Time Machine 2) Reinstall Tahoe (no way - system shows it would take me back to the first Tahoe) 3) Web Browser and 4) Disk Utility and I'm getting ready to look at #4. (I have the silcon M4 chip.)

So coming up in safe mode doesn't seem to be available. It's making me pretty nuts and if I can find a safe place to take my laptop, it'll have to go in to get unscrambled.

You started in RecoveryOS, not safe mode. Follow these instructions carefully:

Start up your Mac in safe mode - Apple Support

1. On your Mac, choose Apple menu > Shut Down.
2. Wait for your Mac to shut down completely. A Mac is completely shut down when the screen is black and any lights are off.
3. Press and hold the power button on your Mac until “Loading startup options” appears.
4. Select a volume.
5. Press and hold the Shift key, then click Continue in Safe Mode.
6. The computer restarts automatically. When the login window appears, you should see “Safe Boot” in the menu bar.

If you don't see the button saying Continue in Safe Mode, then you won't boot in Safe Mode.

Have people had success coming up in Safe Mode?

Yes, the identical problem had been encountered in the past so try it too.

Oh that's not Safe Mode. Refer to the instructions and try again.

Apple likes to be concise, but their Safe Mode instructions could benefit from a few more details. The following is the way it's supposed to work, expanded to include those additional details:

1. Start with a Mac powered on in the usual manner, logged into your User Account.
2. Shut down the Mac using  (Apple menu) > Shut Down...
3. Wait until the Mac actually shuts down. You can confirm it has actually shut down by pressing the Caps Lock key. The Mac will begin to start, but the Caps Lock LED will not illuminate until after the Apple logo appears. If that did not happen then it was not shut down, so you might as well start over at Step 1.
4. Press and hold the power button / Touch ID button. "Continue holding for startup options..." will appear, followed a second or two later by "Loading startup options...". If that does not happen, you might as well start over from the beginning.
5. When the "hard disk" and "gear" icons appear, release the power button.
6. Select (momentarily click) the startup disk. "Continue" will appear beneath it.
7. Press and hold either Shift key. The "Continue" button legend should change to "Continue in Safe Mode".
8. Continue holding the Shift key and click that button. You can release the Shift key when the indeterminate progress indicator "spinner" appears.
9. The usual Login screen will appear, but "Safe Boot" should appear in red at the upper right of the Mac's menu bar.
10. Log in in the usual manner. Once logged in, it might not be obvious the Mac is in Safe Mode. You can confirm that it is by holding a Control key when clicking  > System Information... > Software. Under System Software Overview, the words Boot Mode: Safe should appear. Get system information for your Mac - Apple Support describes how to open the System Information app.

Ten steps instead of four, but it should be abundantly clear about what you're doing and what you should expect.

Edit: even more steps than neuroanatomist's 😄

If you are near an Apple Store, consider making a Genius Bar appointment.

https://www.apple.com/retail/geniusbar/

It is not a problem to upgrade from an older version of macOS. If you are running any version of Tahoe (or older) on your Mac, and your Mac is compatible with Golden Gate, then you will be offered 27.2 by Software Update.

It was unclear from your first reply, is Low Data Mode OFF? You will not see the update if that setting is on.

Regarding the Terminal command, when I paste the command from my earlier post into the command line I see Scanning for 27.2 installer for a few seconds, then Installing with a percent complete counter that starts incrementing up: