Jessica

increasingly convinced brew cask should not ever be used

Linuxbrew doesn't really like being installed on a Synology NAS, you have to force the issue in a few important ways.

1. Create a /usr/bin/ldd shim so the Homebrew installer can figure out it's compatible with the system glibc: https://community.synology.com/enu/forum/1/post/153781

   #!/bin/bash
   [[ $(/usr/lib/libc.so.6) =~ version\ ([0-9]\.[0-9]+) ]] && echo "ldd ${BASH_REMATCH[1]}"

   made with @nex3's syntax highlighter
2. Mount /volume1/homes to /home:
   sudo mkdir /home && sudo mount -o bind "/volume1/homes" /home
3. Install Homebrew using the normal curl-based instructions on https://brew.sh
4. Download the latest amd64 static curl Linux binary to ~/bin/curl:
   https://github.com/moparisthebest/static-curl/releases/latest
5. Override curl detection in $(brew --prefix)/Library/Homebrew/brew.sh):
   In setup_curl function, replace the literal HOMEBREW_CURL="curl" with HOMEBREW_CURL="${HOME}/bin/curl
6. Install a compiler from Homebrew:
   brew install gcc
7. Force Linuxbrew to find its own copy of gcc:
   ln -s $(which gcc-12) /home/linuxbrew/.linuxbrew/bin/gcc
8. Install curl from Homebrew:
   brew install curl

Now you have a working base setup which can install things. You can also revert the curl detection. Good luck!