Well, after a lot of searching, and asking for help, I found something. Maybe this can help someone else.
I still do not understand what the problem is, but I have a workaround.
Since Android studio was unable to install the SDK, I used homebrew to install android-sdk separately and then linked this folder as ANDROID_HOME and then set this path to the Android Studio SDK, and it worked.
There were problems in updating the SDK that I ran into, but updating from the command line did the trick.
Now everything is working fine.
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