fix(staged): run doctor checks with shell env snapshots

[matt2e]

Summary

• add doctor environment snapshots for binary resolution, checks, freshness probes, and fix execution
• capture and sanitize Staged home shell env with an extended PATH for doctor subprocesses
• route Tauri and web doctor entry points through the snapshot-aware APIs

[chatgpt-codex-connector]

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[chatgpt-codex-connector]

Preserve user npm prefix env

When a user's shell intentionally exports NPM_CONFIG_PREFIX to a user-writable global prefix (for example ~/.npm-global), this sanitizer drops it before run_doctor_fix/run_doctor_update executes npm install -g .... I confirmed npm treats that env var as the global prefix (NPM_CONFIG_PREFIX=/tmp/x npm prefix -g returns /tmp/x), so these doctor fixes/updates will run against npm's default prefix instead, often failing due to permissions or installing a duplicate outside the PATH that doctor just resolved.

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