chore(start): migrate src/start from JavaScript to TypeScript

[JustAGhosT]

Summary

The /start TUI (src/start/) is currently written in plain JavaScript (.jsx, .js). It should be migrated to TypeScript for consistency with the project's type-safety rules.

Motivation

• All new code follows strict: true TypeScript per .claude/rules/languages/typescript.md
• src/start/ was bootstrapped quickly with Ink/React JSX before the TS rule was enforced
• Type errors are currently caught by JSDoc @typedef annotations in some files but not others
• Moving to .tsx/.ts enables IDE autocomplete, cross-file refactoring, and compile-time bug detection

Scope

src/start/components/*.jsx  → *.tsx
src/start/lib/*.js          → *.ts
src/start/index.js          → index.ts

Add tsconfig.json for src/start/ (or extend root tsconfig).
Update vitest.config.* to handle .tsx/.ts imports.

Acceptance Criteria

• All .jsx → .tsx with explicit prop types (no any)
• All .js lib files → .ts with explicit return types
• strict: true — zero type errors
• All 156 existing tests pass without modification
• CI passes (lint + typecheck + tests)

Related

• JSDoc typedefs to promote: src/start/lib/worktrees.js:14, src/start/lib/tasks.js:13
• Coverage must not decrease (qa-coverage-threshold rule)

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