Even though you don't have a backup, you still may have some options to attempt data recovery from your SSD as long as the SSD has not been erased or the data deleted.
You can try putting the MBAir into Target Disk Mode to connect the laptop to your newer Mac so the MBAir will appear as an external drive. You will need an Apple USB-C to Thunderbolt 2 Adapter along with an appropriate older style Thunderbolt cable to connect the adapter to the older MBAir.
As long as your MBAir is using the original Apple SSD, then you can also remove the SSD and install it into an OWC Envoy Pro Enclosure so you can connect the SSD to your newer Mac. You may need to also use an Apple USB-C to USB-A Adapter depending on the connections of this enclosure.
If your laptop has a third party M.2 SSD with adapter installed, then you can use a standard USB to M.2 SSD Adapater/Enclosure, but if the third party SSD is an NVMe SSD such as the OWC Aura Pro, then there are no adapters/enclosures for that SSD.
When trying to access the data on an external drive using either of these two methods it can sometimes be difficult to locate the mount point for the User folder containing your data as macOS may mount it in a couple different places depending on the versions of macOS involved.
If you cannot access your data using these methods, then you should try reinstalling macOS over top of itself making sure to select the "Macintosh HD" volume for the destination of the install. Do not select the "Data" volume or "Macintosh HD - Data" volume or you will end up with more work to do and possibly run out of storage space. "Macintosh HD" is the default volume name used by macOS unless you changed it at some point. As long as you do not erase the SSD or any volumes, then if your data is there, it should still be "safe".
Your laptop should be able to boot into Internet Recovery Mode using Command + Option + R to access the latest online macOS installer for your laptop. However, you do need a good Internet & local network connection plus sometimes Internet Recovery Mode will only boot to the online installer which originally shipped with the laptop from the factory which in this case won't help you to recover the data as it would require erasing the whole physical drive to install macOS 10.11.
You can also try installing macOS to an external USB drive and boot from the external USB drive on your MBAir to attempt to access your data on the internal drive. Just make sure to erase the external drive properly and point the installer to the external drive or you may erase the data on your internal drive.
To use your 2019 laptop to create a bootable macOS USB installer, you can use the instructions in this Apple article:
How to create a bootable installer for macOS - Apple Support
The quality of USB sticks are extremely poor, so you should try using another USB stick especially a different brand. Also Macs can be very picky about the USB sticks used for booting.
