MCP server and tool handlers should be split into smaller focused modules

[Widthdom]

Found while dogfooding the current source with cdidx symbols and hotspots.

src/CodeIndex/Mcp/McpServer.cs and src/CodeIndex/Mcp/McpToolHandlers.cs remain large, high-reference files. McpToolHandlers.cs is around 5k lines, and the main server file is around 4k lines.

This makes it harder to audit request dispatch, cancellation, error sanitization, indexing, and tool-specific behavior independently.

Expected: split MCP request loop, dispatch/cancellation, error/log formatting, and individual tool handlers into narrower modules with focused tests.

[Widthdom]

Work started on branch codex/fix-issue3382-3545-3522-3405.