This is extraordinary - not in a good way!
I had this problem on another Catalina machine, about a month ago, and i went back and forth between "xcode-select --install" and deletions. Eventually, i have no idea what i did, CLT was installed. Hura. I thought, "this will never happen again" because i didn't realize it was a widespread phenomenon, and forgot about it.
Now i'm trying to get brew to work on another Catalina machine. Whatever my fooling around was before i am failing to "just do it". So, i followed the instructions here... i went to the Apple Developer site and downloaded the CLT and now brew works again.
But, here's the thing... the downloaded .dmg is ~435MB, and the installer used >2.4GB disk space! What?! I only want the CLT. If i wanted a package that big i would install the whole Xcode package and just hope that the client doesn't do anything destructive by mistake.
I really don't like Apple's way of using our machines the way THEY want, not the way we want. I have already transitioned myself to a linux box, and i'm seriously thinking of moving my clients with me.
Solution found, but Apple's solution is not appreciated.
ps Just to be clear, i appreciate the community for finding and publishing the solution. I just think Apple does not care about small Developers.