ci(branch-checks): run Rust checks in Nix shells

[SDAChess]

Summary

Run Rust branch checks through the flake’s default development shell, keeping CI toolchains and native dependencies consistent with local development.

Related Issue

No issue required: localized CI workflow refactor.

Changes

• Run Rust checks on x86_64-linux, aarch64-linux, and aarch64-darwin.
• Replace the CI container, mise, and sccache with Nix and Cachix.
• Retain rust-cache for Cargo registry and build artifacts.
• Remove brittle telemetry and CA-root dependency checks.

Testing

• mise run pre-commit passes
• Unit tests not required for this CI-only change
• E2E tests not required
• Workflow passes actionlint validation

Checklist

• Follows Conventional Commits
• Commits are signed off (DCO)
• Architecture docs not required