In case anyone is more recently searching the internet, as to how to get rid of the red update badge in Big Sur (especially for those of us stuck on 11.2.3 - because, until a solution is found, certain Mac models [such as my Mac Pro 5,1] currently can't install or regularly or reliably boot 11.3.x, 11.4.x or 11.5beta without KPs, stalls or other significant problems, because of apparent 'race conditions' addressing PCIe slots during the boot process), then please read on...
The 'defaults delete com.apple.systempreferences AttentionPrefBundleIDs; killall Dock' method is only temporary in Big Sur. It hides the red update badge after the Terminal command, but the distracting badge reappears again after a reboot.
For me, the best solution was to find the location of the red circular badge files and alter or replace them with transparent versions. See the screen shots below for the results on my computer.
The two file locations are as follows (in each case, replace 'BigSurSystemVolume' with the name of your system volume):
(1) to remove the System Preferences
Dock red circular badge, leaving just the inconspicuous '1' behind (which I can happily live with until I 'can' update past 11.2.3), alter this file: '/BigSurSystemVolume/System/Library/CoreServices/Dock.app/Contents/Resources/statuslabel.png'. Make a copy of the original file and open it with a graphics editing app. Simply delete the red circle, leaving just the transparent alpha layer. Save the file somewhere and replace the original one by either dragging it into the Resources folder (needs to be done while booting into 10.12, 10.13 or 10.14) or using 'mv' with the Terminal from Big Sur recovery (you can't replace a Big Sur system volume file when booting normally into that OS - you'll get the infamous 'prohibited circle').
(2) to remove the System Preferences
Software Update red circular badge, leaving just the '1' behind, alter this file: '/BigSurSystemVolume/System/Applications/System Preferences/Contents/Resources/Assets.car'. The original Assets.car file has 2 red badge icons contained inside. Use ThemeEngine.app* to open the file. Drag the 2 icons to the desktop and edit them with a graphics app. Again, simply delete the red circles, leaving just the transparent alpha layer. Save the dragged-out files and drag them back over the top of the 2 original icons inside the open Assets.car file. Note - one icon is for regular (non-retina) displays and the other, the '2x' one, is for retina displays. Save the Assets.car file and either drag it over the original one (from 10.12, 10.13 or 10.14) or using 'mv' from Big Sur recovery.
If using the 'mv' move command, there should be no need to apply a new BS snapshot or rebuild the kext collection on the Preboot volume. You are only replacing internal app files, not root-level system files.
*ThemeEngine is available for download from various sites:
The original version works up to/incl. 10.14 -
https://github.com/alexzielenski/ThemeEngine/releases/tag/1.0.0(111)
A version for 10.15 (compiled by Allan Nyholm) -
https://www.dropbox.com/s/911ao8l1g480c7a/ThemeEngine 2 for 10.15.zip?dl=0
A recent port by Jeremy Legendre for 11.0.1 upwards -
https://github.com/jslegendre/ThemeEngine