Yes you should not have erased your mac and then gone on to rely on reinstalling
via Apples Recovery Servers because of the certificate issue.
Ideally you should have downloaded a copy of El Capitan which is still available
from Apple (with valid certificates) from here, How to get old versions of macOS – Apple Support (UK)
Go to the Downloads section and click on Mac OS X El Capitan.
This would have downloaded InstallMacOSX.dmg which you open to get InstallMacOSX.pkg which you open
to create the Install OS X El Capitan.app in the Applications folder.
With the Install OS X El Capitan.app and the createinstallmedia command in Terminal you can create a bootable
USB flash drive installer, How to create a bootable installer for macOS – Apple Support (UK)
You would then boot to the installer USB, erase the drive with Disk Utility and reinstall Mac OS X El Capitan.
If you have access to another mac that you know is capable of running El Capitan then you could still do this
and then transfer the bootable USB to your MacBook Air and reinstall that way.
It is important that the donor mac is capable of running El Capitan (it may nave been upgraded since,
but that is OK) any mac that was bought brand new with a newer OS (Sierra onwards) will refuse the process
where it creates the Install OS X El Capitan.app from the InstallOSX.pkg.
If you have or can borrow a mac to do the above then I can give you more detailed instructions on what to do.
Did you make a full backup of your mac before you erased the drive if so you could reinstall El Capitan
that way using the Restore from Time Machine.
If you can only reinstall from Apples Recovery Servers then you may have to get that Thunderbolt to
Ethernet adaptor, unless of course you know someone you could borrow one from. Relying on WiFi
will reset the date back to today and you will encounter the earlier issue.
