VS Code on macOS will not auto update if it’s running from a read-only folder.
If you downloaded and are running VS Code from your Downloads folder, which is read-only, it will not auto update.
Solution:
When you’ve downloaded VS Code, drag the downloaded file to your applications directory and run it from there instead.
It will now auto update
Thanks to @gino mempin in the answer to my previous question for the inspiration How to turn on auto search on VS Code on macOs?