You seem to have a reasonable grasp on the basics, at the very least (and possibly much more), so I'm going to try to give you a "partial answer" that might get you in the right direction. Normally I'd try to test things out more fully on my end before posting, but that could take several days.
I have a feeling that you are running into similar issues as we're seeing with camera support in WSL2. While the WSL2 kernel has USB/IP support, it doesn't contain drivers for most USB devices themselves. For instance, looking at the WSL2 kernel config, you'll see that MMC (SD card) support is not included.
I can't guarantee that it will work, but you will at least have to compile your own kernel with MMC support, and likely the actual driver for your SD card reader itself as well.
On a typical Ubuntu (or other distribution) kernel, all of the common hardware devices would be available as kernel modules, but WSL's kernel is more streamlined.
Another, easier alternative, since you are on Windows 11, might be to mount the actual drive from Windows to WSL using wsl --mount --bare
Good news, it is now possible to mount USB media (including formatted as FAT) and network shares with drvfs on Windows 10:
Mount removable media: (e.g. D:)
$ sudo mkdir /mnt/d
$ sudo mount -t drvfs D: /mnt/d
To safely unmount
$ sudo umount /mnt/d
You can also mount network shares without smbfs:
$ sudo mount -t drvfs '\\server\share' /mnt/share
You need at least Build 16176 so you might have to opt-in to the Windows Insider programm and then update Windows. Source: https://blogs.msdn.microsoft.com/wsl/2017/04/18/file-system-improvements-to-the-windows-subsystem-for-linux/
I like to use Rufus when performing this type of operation on a Windows machine. This program is great for creating bootable media, particularly when using Linux images. Also, the "DD" function works really well.
This software is free and open source.
If you insist on using your Ubuntu subsystem, type the command "tail -f /var/log/syslog" or "dmesg -w" then connect the mmc. This will display where it has been made available in userspace, and will be a good troubleshooting step should nothing happen. You may not have these commands available depending on how Ubuntu is configured.
Summary
In the latest Windows Insider build [16176], the Windows Subsystem for Linux (WSL) now allows you to manually mount Windows drives using the DrvFs file system. Previously, WSL would automatically mount all fixed NTFS drives when you launch Bash, but there was no support for mounting additional storage like removable drives or network locations.
Now, not only can you manually mount any drives on your system, we’ve also added support for other file systems such as FAT, as well as mounting network locations. This enables you to access any drive, including removable USB sticks or CDs, and any network location you can reach in Windows all from within WSL.
Mounting DrvFs
In order to mount a Windows drive using DrvFs, you can use the regular Linux mount command. For example, to mount a removable drive D: as /mnt/d directory, run the following commands:
$ sudo mkdir /mnt/d $ sudo mount -t drvfs D: /mnt/dNow, you will be able to access the files of your D: drive under /mnt/d. When you wish to unmount the drive, for example so you can safely remove it, run the following command:
$ sudo umount /mnt/d
Source: File System Improvements to the Windows Subsystem for Linux