Feature Request: Remote Control support for VS Code chat mode

Feature Request

Problem

Remote Control currently only works with terminal-based Claude Code sessions. Users who prefer the VS Code chat mode (Claude Code extension) cannot use Remote Control to continue their sessions from a mobile device or browser.

Proposed Solution

Extend Remote Control functionality to support the VS Code integrated chat mode, not just terminal sessions. This would allow users to:

• Start a coding session in VS Code chat mode
• Continue managing it from their phone via claude.ai/code or the Claude mobile app
• Maintain access to all local MCP servers, environment variables, and project configurations

Why This Matters

Many users prefer the VS Code chat experience over the terminal for its better UI integration, inline diffs, and seamless editor workflow. Limiting Remote Control to terminal-only forces these users to either:

1. Switch to terminal mode (losing their preferred workflow)
2. Use third-party remote desktop solutions (suboptimal experience)

Additional Context

Remote Control is a great feature — extending it to VS Code chat mode would make it even more powerful and accessible to a larger portion of the user base.