Aha! No, you don't need to use Disk Utility. The main drive is already mounted. All the user has to do is type in the correct volume name, as you show for the Preferences folder. So, here's the whole thing again with the correct steps.
Start up in Recovery Mode. Intel method. Apple Silicon method.
On an Intel Mac, all you need to do restart and then immediately hold down the Command+R keys. Then enter your admin password if prompted.
On an Apple Silicon Mac, shut the Mac down. Then press and hold the power button until you get a screen showing any bootable volumes and Options. Double click the Options icon. Next, select the admin icon and click Next. Enter the admin password. You can't jump directly to desktop folder with ~/Desktop.
So, once at the Recovery Mode work screen, go to the top menu bar and launch Terminal. Type this (case doesn't matter here):
cd /volumes/drive_name/users/your_account/desktop
Examples: If your main drive has a space in the name, you have to use a backspace to parse the space character. So, a boot volume named My Mac would be:
cd /volumes/my\ mac/users/your_account/desktop
Mine is simply Sonoma, and your short account name is normally always one word. An admin of Jim Smith would then be jimsmith for the account name. Then the command would be:
cd /volumes/sonoma/users/jimsmith/desktop
To verify you're in the correct place, type this:
ls
That's a lower case L, not a capital i. It's short for list. You should see a very long list of files scroll by. If you do not see the desktop files, back up to the root of the drive with:
cd /
Start over. It may be easier to do this one step at a time since you can check each move. Like this:
cd /volumes/drive_name
ls
You should see everything at the root of the drive. Applications, Library, System and Users.
cd /users
ls
You should see Shared and all user account names.
cd /your_account
ls should then show Desktop, Documents, Downloads, Public and other typical folders.
cd /desktop
ls
Once you've verified you've seen thousands of image file names scroll by, you delete all of them in one command. What you type depends on what kind of file they are.
Make absolutely sure to NEVER type *.* . That means everything.
(in each example below, there's a space between rm and the file type)
You would use this command to remove all JPEG images ending in .jpg
rm *.jpg
If the file extension is .jpeg, then it would be:
rm *.jpeg
This for Apple HEIC images
rm *.heic
You can check what's left after a command by typing ls again.
When the images have all been removed, you can quit Terminal and restart the Mac normally. Or, while still in Terminal, simply enter reboot.