MacBook Air 2017 doesn't recognize original SSD

If your laptop had Big Sur installed then the laptop should be at the most recent firmware. The firmware is only able to be updated when upgrading macOS to a new version or with any firmware updates included with any macOS patches/updates. While there may be other methods they are not approved by Apple who will remove any posts discussing it (plus it is risky & beyond the abilities of the average user).

If the laptop worked fine after you had upgraded to Big Sur and the issue only started after you tried installing a third party SSD it would imply the firmware is Ok and that something happened during the attempted SSD upgrade. Possibilities are the new adapter or SSD damaged the Logic Board, or you accidentally "zapped" the Logic Board or both SSDs with static electricity, or accidentally damaged a very small component on the Logic Board, or you did not unplug the charger & battery before performing the upgrade there by damaging the Logic Board and SSDs. There are a lot of ways something unintentionally occurred and it is always a possibility no matter how careful a person is.

Did you receive that diagnostic error code with the original Apple SSD installed? It is known that the Apple Diagnostics will report a failure when a third party SSD (or other third party parts) are installed.

Did you try using a different brand of USB stick for your macOS installer? Macs can be very picky about the USB sticks used for booting. You can even use a standard USB hard drive/SSD for the macOS USB installer.

Install your new third party SSD and try installing Ubuntu or Linux Mint on the laptop. This may give us a better idea of the problem. Sometimes Linux will work where macOS fails. If Linux is unable to boot or install on this laptop, then you have some sort of hardware issue.

Since you are using a standard M.2 SSD you can test the SSD in a standard PC or use an external USB NVMe enclosure to make sure the new SSD is functional which will narrow down the problem to the SSD adapter or the Logic Board. Of course this is a risky test if the SSD is bad because it may damage another motherboard or the enclosure.

You can test the original Apple SSD by installing it into an OWC Envoy Pro Enclosure and trying to access the contents using another Mac.

If the Logic Board or firmware is bad, then your only options are to have Apple replace the Logic Board (expensive), or to find a repair technician that knows how to repair an Apple Logic Board and/or firmware, or you recycle this system and buy another computer. It is also possible that the Big Sur firmware has a bug which is causing your problems (the Big Sur installer has been reported to have damaged some 2013 & 2014 MBPros during the install process).

Make sure the contacts of the original Apple SSD are all clean & shiny. Use compressed air to blow out any dust that may be inside the SSD connector slot (make sure to keep the can of compressed air completely upright or you can damage the Logic Board with the freezing cold wet propellant).

What version of macOS was installed on this laptop? macOS 10.13+ must be installed before installing a third party SSD since the laptop's firmware must have a firmware update that will allow a third party NVMe SSD to be recognized. After the third party SSD is installed you must use macOS 10.13+ to see the third party SSD as the necessary NVMe driver only became available starting with macOS 10.13.

Using a standard M.2 SSD in this laptop requires the use of a third party adapter as well. There is always a possibility this third party adapter is defective and damaged your Logic Board.

With the original Apple SSD installed boot into Internet Recovery Mode (Command + Option + R) or from a bootable macOS USB installer (for Big Sur) to see if you can have Disk Utility repair the file system on the Apple SSD. If the file system is good, then try reinstalling macOS over top of itself.

I hope you have a backup of your data. You should always have frequent & regular backups since it is impossible to recover accidentally deleted data from an SSD plus an SSD can fail at any time without any warning signs.

Try running the Apple Diagnostics with the original Apple SSD installed.

I've been running that VDH002 error code through Google and stumbled upon these :

https://support.apple.com/en-us/HT203747

https://nl.ifixit.com/Answers/View/559054/Random+freezes+&+VDH002+%22There+may+be+an+issue+with+a+storage+device%22

It would seem that you may have damaged your Mac.