When you say "When I got to the Apple Store and try to download the installer, it simply will not.", do you mean that when you go to App Store you cannot find the macOS 11 Big Sur installer?
If so, that is normal: Big Sur is no longer the current version of macOS, and App Store only gives easy access to the current version (which is currently macOS 12 Monterey).
To download the full installer for an older macOS version you need to use an indirect method. Apple has a support article which provides links for each available version.
For macOS 10.13 High Sierra and later, the link takes you to a hidden page on App Store from which you can get the installer. It will end up in your Applications folder, named along the lines of "Install macOS name.app".
For OS X 10.10 Yosemite through macOS 10.12 Sierra, the link gets you a disk image (.dmg) file in your Downloads folder, which contains an installer package. Running that installer package creates "Install macOS name.app" or "Install OS X name.app" in your Applications folder, after which you can eject the mounted disk image and delete the disk image file.
In both categories, the installer is for the latest minor version of the OS (e.g. Big Sur downloaded now would be 11.6.4, which was released this week).
Note: to get a particular macOS version, your Mac model must be supported by that version. If you need an older or newer version, the workaround is to create a virtual machine for a macOS version which supports your Mac model, then download an older or newer version inside the virtual machine, and move the installer back to the host to use it for creating a virtual machine.
Command line method
If you are running macOS 10.15 Catalina or later, there is also a command line method to download the installer for recent macOS versions:
softwareupdate --fetch-full-installer --full-installer-version 11.6.4
(specify the version number at the end).
If you are running macOS 11 Big Sur or later there is also a way to find out which versions of the installer are available via the softwareupdate tool (it only lists available versions supported by your Mac):
softwareupdate --list-full-installers
Recently, Apple has made a wider range of minor macOS versions available via the command line method. For example, my 2019 16-inch MacBook Pro (which originally shipped with 10.15.1, can't run anything older, and is currently running 12.2.1) lists these versions:
% softwareupdate --list-full-installers
Finding available software
Software Update found the following full installers:
* Title: macOS Monterey, Version: 12.2.1, Size: 12155426708K, Build: 21D62
* Title: macOS Monterey, Version: 12.2, Size: 12156325205K, Build: 21D49
* Title: macOS Monterey, Version: 12.1, Size: 12157035487K, Build: 21C52
* Title: macOS Monterey, Version: 12.0.1, Size: 12128428704K, Build: 21A559
* Title: macOS Big Sur, Version: 11.6.4, Size: 12439328867K, Build: 20G417
* Title: macOS Big Sur, Version: 11.6.3, Size: 12435122667K, Build: 20G415
* Title: macOS Big Sur, Version: 11.6.2, Size: 12433351292K, Build: 20G314
* Title: macOS Big Sur, Version: 11.6.1, Size: 12428472512K, Build: 20G224
* Title: macOS Big Sur, Version: 11.5.2, Size: 12440916552K, Build: 20G95
* Title: macOS Catalina, Version: 10.15.7, Size: 8248985973K, Build: 19H15
* Title: macOS Catalina, Version: 10.15.7, Size: 8248854894K, Build: 19H2
* Title: macOS Catalina, Version: 10.15.6, Size: 8248781171K, Build: 19G2021
All minor versions of macOS Monterey are available, but only late minor versions of macOS Big Sur and macOS Catalina.
I've found that the latest version sometimes doesn't appear in the list, e.g. 11.6.4 wasn't shown until I actually downloaded that installer, or tried to download a nonexistent future version.
For comparison: a 2015 MacBook Air running macOS Big Sur also lists 10.14.6 and 10.13.6, because those versions can run on this Mac. (macOS 12 Sierra and earlier have never been available via this method.)
