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Oh dear, I think I see your problem. Unfortunately you may need to perform an erase and install from a different source. The recovery partition can only do so much. What you need is a Bootable installer. This would usually be in the form of a thumb drive or a a Bootable Clone backup. This can be easily done at an Apple Genius bar or Service Centre. Unfortunately you cannot do this yourself because you cannot download the installer after erasing the disk.
 
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All models of the air from mid-2013 to 2017 use the same ssd.

Nvme drives will work but there are some minor issues: https://forums.macrumors.com/threads/upgrading-2013-2014-macbook-pro-ssd-to-m-2-nvme.2034976/

I have an Intel 600p nvme in an air early 2015 that works well and was a lot cheaper than getting a used apple ssd.

All the pics I looked up for the Intel 600p have a different slot configuration. Are you sure about compatibilty? The 2014 on Airs have a slot closer to the middle of the pins, not at the edge like the one you are talking about.

Intel 600p and then the 10215 Apple SSD

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In order to use an nvme drive in the MBA, you need to use a pci-e adapter like the one from Sintech which is why I included a link in my post to the nvme thread on Macrumors.

I should have been clearer by saying that all MBAs from mid-2013 to 2017 use the same Apple oem SSD.
 
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