Apple retail has installed Catalina upon my visit last week but since Big Sur isn't available on the app store, they couldn't install it.
They didn't do you any favors.
I understand you now have a Mac that won't boot and you're dead in the water. I think your only option at this point is to boot Recovery again and completely erase the internal storage. Then, reinstall that Mac's originally installed operating system, create a User Account with a throwaway name, upgrade its version of macOS to Big Sur, and finally migrate its backed up content from your Big Sur Time Machine backup. Lastly, remove the temporary User Account with the throwaway name.
Here are the relevant steps.
Begin with Erase and reformat a storage device in Disk Utility on Mac - Apple Support.
Drawing your attention to its Step 2, you need to restart your Mac in Recovery mode as it describes. It will lead you to Use macOS Recovery on an Intel-based Mac - Apple Support, in which it describes several ways to boot Recovery. You will want to follow this third bullet point under Start up your Intel-based Mac in macOS Recovery:
- Option-Shift-Command-R: Start up from macOS Recovery over the internet. Use this key combination to reinstall the version of macOS that came with your Mac or the closest version that’s still available.
I do not know what that version will be, but it's not important because you will be upgrading it later.
Then, returning to Erase and reformat a storage device in Disk Utility on Mac - Apple Support, draw your attention to its Steps 3 and 4. In that context, "the device" refers to the entire storage device, not a partition or volume subservient to it. It will be the icon at the extreme upper-left corner of the device list.
- This is necessary because your MBA would have had its internal storage device reformatted using APFS sometime prior to upgrading it to Big Sur — a format that its originally installed version of macOS will not recognize. If you don't erase the entire device, then the installer will not be able to find the device on which to install macOS.
Then, proceed to install macOS: quit Disk Utility, return to the main macOS Utilities screen, and select Reinstall macOS. Follow the prompts to select the MBA's internal storage device, which will hopefully be present. Take a break for a while and let it finish.
When it finishes, it will ask if you want to transfer content from another Mac or its Time Machine backup. Decline that option. When asked to create a User Account, provide a throwaway name.
Log in to that Account, then proceed to upgrade that Mac to Big Sur. No, it's not in the App Store. Instead, follow these instructions: How to download macOS - Apple Support. Scroll down to Big Sur 11. That's what the Apple Store should have done.
Take another break. In the end, your MBA will be running Big Sur, but it won't have any of your User Account or Applications or anything else that you had prior to the botched upgrade.
Then, use Migration Assistant to migrate your Time Machine backup of your usual User Account. Follow these instructions: Transfer your information to Mac from another computer or device - Apple Support.
Take another break. When it eventually finishes, you will have two User Accounts. The first, temporary one, and the one you will use as your usual Account. The temporary Account can be removed according to Delete a user or group on Mac - Apple Support but there is no pressing need to do that right away.
After that, at your option, you can decide if you want to upgrade to Monterey, which is where you began.
Summary: It is nearly certain that some of the above will not go exactly as I wrote it. I probably forgot a step or two, but that's essentially what you're up against since your Mac is essentially useless at this point. Please write back if something doesn't make sense of if you get stuck somewhere.
