Does Homebrew work on Linux

Yes. Homebrew supports Linux (Linuxbrew). Quick setup:
/bin/bash -c "$(curl -fsSL https://raw.githubusercontent.com/Homebrew/install/HEAD/install.sh)"
echo 'eval "$(/home/linuxbrew/.linuxbrew/bin/brew shellenv)"' >> ~/.profile
eval "$(/home/linuxbrew/.linuxbrew/bin/brew shellenv)"
brew install <formula>
If you run OpenClaw via systemd, ensure the service PATH includes /home/linuxbrew/.linuxbrew/bin (or your brew prefix) so brew-installed tools resolve in non‑login shells. Recent builds also prepend common user bin dirs on Linux systemd services (for example ~/.local/bin, ~/.npm-global/bin, ~/.local/share/pnpm, ~/.bun/bin) and honor PNPM_HOME, NPM_CONFIG_PREFIX, BUN_INSTALL, VOLTA_HOME, ASDF_DATA_DIR, NVM_DIR, and FNM_DIR when set.