Why does brew say that nodejs is installed?

First of all, I run node in macOS terminal, and get

bash: /usr/local/bin/node: No such file or directory    

I run brew install node and get

Warning: node 12.10.0 is already installed, it's just not linked You can use `brew link node` to link this version.

So I run brew link node and get

Error: Could not symlink include/node/common.gypi                                                                 
/usr/local/include/node is not writable.   

I suppose that's some permission issue. So I run it with sudo and get

Error: Running Homebrew as root is extremely dangerous and no longer supported.                                   
As Homebrew does not drop privileges on installation you would be giving all                                      
build scripts full access to your system. 

Then I try brew uninstall node, and repeat all the above steps, and still get the same errors. What do I do?

from Why does brew say that nodejs is installed?