the Macbook air is unable to update its os beyond 10.13.6 and neither there is any avail sign to update further! uncanny upon the prior question asked by the user and seemed there wasn't any answer or solution by apple delegates

The macOS version you can upgrade to depends on the year your Mac was released.

Click the  (Apple) menu at the top left of your screen and choose About This Mac. 

macOS 11 Big Sur is compatible with the Macs listed here:

https://support.apple.com/en-us/HT211238

macOS 10.15 Catalina is compatible with the Macs listed here:

https://support.apple.com/en-us/HT210222

macOS 10.14 Mojave is compatible with the Macs listed here:

https://support.apple.com/kb/SP777?locale=en_US

Once your Air has installed macOS 10.13.6, the only other updates that would have appeared in the Updates tab of the Mac App Store would have been Security updates. High Sierra has now been unsupported by Apple for nearly a year, and there will be no further updates for it.

The age of your Air will determine what newer operating system you may be able to upgrade too. From menu > About this Mac, look at the parenthesis content on the product description line. Keep that in mind when you use Safari to visit the Apple How to get old versions of macOS site. There you compare that parenthesis data with the hardware requirements of newer operating systems, to determine what your upgrade options are.

For instance, your Air must be a mid-2012 model or later to upgrade to macOS 10.14.6 (Mojave).

Which MacBook Air model year? It's the Mac model year that determines your available upgrades.

How to get old versions of macOS - Apple Support

How to upgrade to macOS Big Sur - Apple Support

Monterey is just out. If you're still on something as far back as 10.13, I'd probably give Monterey few weeks to sort itself out.

Have a complete and current backup created before the upgrade, as there is no downgrade available.

If your MacBook Air supports Catalina or newer, you'll need to have all of your apps 64-bit, and upgrade or update or replace or retire those that are 32-bit. This includes Microsoft Office 2011 and earlier. You'll need Office 2016 (deprecated), Office 2019 (current), Office 2021 (soon), or an Office 365 subscription, or you'll need a replacement such as LibreOffice (free), or the Apple iWork apps that can read and write many Office documents. You can use the Go64 tool to find existing older 32-bit apps, if your MacBook Air supports Catalina or later and you're upgrading to Catalina or later.

And did I mention having a complete backup before the upgrade?