I think it's actually a combination of both:
You have to have Xcode installed and Command Line Tools only get installed if downloaded manually from the page linked above.
At least that was the case for me two times now.
We used to be able to install Command Line Tools standalone without the need for the whole Xcode which personally i don't need.
As Homebrew bottles depend on Command Line Tools for installation i hereby plea to reinstate that functionality.
Zap42
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