It's important to understand that there are multiple components involved in a WSL installation, and whether you can "move" or change their installation location differs for each of those components:
tl;dr Most users are going to want to move the largest consumer of disk space - The WSL distribution itself. That is Item #5 below (read it for prerequisites), but short-version on the latest release:
wsl --manage <distro_name> --move <new_location>
Much more detail:
The wsl.exe command, which is what you mainly appear to be asking about, is built-in to Windows in recent releases. You didn't (and can't) actually install or uninstall the actual wsl.exe through any of the Windows Features or Apps & Features settings. It is present when you install Windows.
The base Windows Subsystem for Linux ("lxss", to some degree), which is also built into Windows as a "Windows Feature". As a Windows Feature, it can be enabled or disabled, but not, to my knowledge, uninstalled. As you can see in my post here, this takes up very little disk space.
The WSL2 Virtual Machine Platform, which is also a Windows feature.
Like all other Windows features, no, you can't change the location where the files live. Most will be under System32. The fact that you were in that directory when you enabled the feature has nothing to do with it.
I can't recall which of the files in the C:\Windows\System32\lxss directory are there when Windows is installed.
The other components, which are not built-in to Windows, are:
The WSL2 Linux kernel. This is actually what was removed when you uninstalled the "Windows Subsystem for Linux Update". I agree that it is confusingly named.
This official kernel is also installed in a subdirectory with the other WSL binaries, in C:\Windows\System32\lxss\tools. While you can't change the folder where this kernel is installed, you can move it to another directory if you'd like and create a file in your Windows (not Linux) user profile directory named .wslconfig with the following contents:
[wsl32]
kernel="C:\path\to\kernel"
You can also compile additional kernels and place them whereever you like, pointing to the one you want to launch in the .wslconfig.
One or more WSL distributions.
When you install a distribution using wsl --install -d <distro> or from the Microsoft Store, it is installed by default in %USERPROFILE%\AppData\Local\Packages\<PackageName>.
These can be "moved" in one of two ways:
Latest releases
As of WSL 2.3.11 (currently pre-release), WSL now includes a built-in command-line option to move a distribution to a new location:
Assuming you are on the "App" version of WSL (most installations in the last 2 years), you can update with:
wsl --update --pre-release
You can then move any WSL2 (not WSL1) distribution using:
wsl --manage <distro_name> --move <new_location>
Older releases
On older releases, you can "move" distributions by exporting them and re-importing them.
From PowerShell:
mkdir D:\WSL\images # For example
mkdir D:\WSL\instances\<newDistroName>
cd D:\WSL
wsl -l -v # Verify distro name to export
wsl --export <distroname> .\images<distroname>.tar
wsl --import <newDistroName> .\instances<newDistroName> .\images<distroname>.tar
wsl --set-default <newDistroName>
You'll also need to set the default username in that copied instance via the /etc/wsl.conf file as discussed in this answer.
It might also be possible to forcibly move the distribution files from your AppData folder to another location and then update the registry corresponding registry location (HKCU:\Software\Microsoft\Windows\CurrentVersion\Lxss). However, I have not tested this, nor seen anyone else try it. If you want to give it a shot, make sure you have a backup via wsl --export above.
