Distribution: Homebrew, npx, and a curl installer

[amrit110]

cargo install backcheck requires a Rust toolchain, which most people who would benefit from
this tool do not have. Release binaries are already built for macOS (arm64/x64), Linux, and
Windows by release.yml — they just need
friendlier front doors.

• Homebrew tap — brew install VectorInstitute/tap/backcheck
• curl | sh installer — detect platform, fetch the right release asset, verify a
  checksum, drop the binary on PATH
• npm wrapper — npx backcheck matters here because a large share of Claude Code users
  live in a JS toolchain and will never run cargo
• checksums + signing on release assets
• cargo binstall metadata
• Nix, Scoop, AUR — nice to have

Why this is worth doing well

The install step is where most people decide whether to try a tool. backcheck is a single
static binary with no runtime dependencies, so every one of these should be a thin wrapper —
there is no reason for the install to be harder than the tool.

The npm wrapper is the highest-value item and a good first issue: a tiny postinstall that
downloads the matching release asset, plus a bin shim.

Please also

Verify the binary works after installing by that route, and add it to the install section of the
README in the same PR.