Like others I was getting no prompt for updating the CLI tools.
Running softwareupdate --install did not work for me.
SOLUTION (for me):
I had to first find the software update manually:
softwareupdate --list
This provided me with the following output:
Finding available software
Software Update found the following new or updated software:
* Label: Command Line Tools for Xcode-13.3
Title: Command Line Tools for Xcode, Version: 13.3, Size: 718145KiB, Recommended: YES,
I then ran:
softwareupdate -i "Command Line Tools for Xcode-13.3"
(NOTE: You pass the "Label:" output as the string as yours could be different.