A1369 Mid-2011 MacBook Air restoration project - clueless though!

The newest macOS version that your 2011 MBA can run is macOS 10.13.6 High Sierra.

I'd suggest you try to create a bootable installer for that OS.

It can still be downloaded and the guidance for creating the bootable installer can be found here: Create a bootable installer for macOS - Apple Support

There is no downloadable ready-made bootable drive, hence the need to "create a bootable installer" using the downloaded file and a USB flash drive of suitable size - minimum 16GB, I believe.

High Sierra has been problematic of late, so if you find yourself stuck at "unable to contact the server" when you attempt to download it, pleae refer to this link: https://mrmacintosh.com/how-to-fix-the-recovery-server-could-not-be-contacted-error-high-sierra-recovery-is-still-online-but-broken/.

The BIG caveat in this process is that you can only create the installer using a Mac that qualifies to actually run High Sierra. You cannot use a modern Mac for that. Qualifying Macs are any number that were released between late-2009/2010 and 2018/2019. If you have a particular Mac in mind for this job, just post back here and someone can tell you if it'll work.

Forget attempting to install Lion on that MBA. I only mention it because you seem to be focused on doing that, and it won't work for a variety of reasons. Short summary: the downloadable Lion installer you found can only be used on Macs that were incapable of running anything later, and yours is capable of running HS. So that's out.

Concentrate on getting High Sierra installed either through Recovery using the tortured workaround D.I. Johnson explained, or by creating a "bootable installer" on an eligible Mac — one that is capable of actually booting and running High Sierra. If I were in your position I'd do the latter. Ask around, borrow someone else's Mac, etc.

That is certainly possible, but the most likely path to success will be to boot Recovery and attempt to install High Sierra from there.

The "MrMacintosh" link and explanation glosses over the fact you have to start with a Mac in Recovery mode — the Internet version of it to be specific — and that's a fundamental prerequisite. If you're seeing the "flashing folder" then you're not in Internet Recovery. It doesn't require a startup disk.

DamianSmith wrote:

Is there a way of building an 'old Mac' VM on a 'new Mac' for exactly this sort of purpose? I suspect Apple would not love any positive answer to this question! I do have parallels and enjoy running Windows 11 nicely on this Mini.

A virtual machine can only run the same basic type of machine code as the host CPU.

Your M2 Mac mini has an Apple Silicon CPU with an ARM-based machine code instruction set. That old 2011 MacBook Air has an Intel CPU. So there is no possibility of running any operating system which the 2011 Mac supports in a virtual machine on the M2 Mac mini.

There is: Parallels Knowledge Base – Run Intel-based virtual machines on Apple silicon Macs using Parallels Desktop x86 emulator, but according to Parallels, it is "still in active development", and is "slow, really slow". There is no indication of support for old versions of macOS.

I would not count on QEMU being able to emulate Intel Mac hardware well enough to even boot macOS.

Plus, in either case, you'd need to prepare a bootable system for the emulator to boot from. If you could do that, you probably would not need the emulator!

I still have a couple of 13" MacBook Pros from 2011, they run High Sierra and work fine although being careful online is paramount because they are not getting security fixes.