[Bug] Chrome MCP tools fail despite `/chrome` reporting connected status

Bug Description
Title: /chrome shows connected but MCP chrome tools fail with "Browser extension is
not connected"

Description:
## Environment

• Claude Code version: [run claude --version]
• OS: macOS (Darwin 25.2.0)
• Chrome extension: installed and shows connected via /chrome

## Steps to Reproduce

1. Start Claude Code CLI
2. Run /chrome - shows "connected"
3. Use any chrome MCP tool (e.g., mcp__claude-in-chrome__tabs_context_mcp)

## Expected Behavior
MCP tools should work since /chrome reports connected status

## Actual Behavior
All chrome MCP tools return:
"Browser extension is not connected. Please ensure the Claude browser extension is
installed and running..."

## Notes

• /chrome command consistently shows connected
• Tried reconnecting via extension - no change
• All chrome MCP tools fail with same error

Environment Info

• Platform: darwin
• Terminal: vscode
• Version: 2.1.19
• Feedback ID: 2d2f56e4-213e-4ea4-aabe-d4c460427ce8

Errors

[{"error":"Error: ENOENT: no such file or directory, scandir '/Library/Application Support/ClaudeCode/.claude/skills'\n    at readdirSync (unknown)\n    at <anonymous> (/$bunfs/root/claude:12:1903)\n    at mW (/$bunfs/root/claude:11:35435)\n    at readdirSync (/$bunfs/root/claude:12:1864)\n    at ILR (/$bunfs/root/claude:2215:380)\n    at ILR (/$bunfs/root/claude:2215:1350)\n    at <anonymous> (/$bunfs/root/claude:2215:4440)\n    at <anonymous> (/$bunfs/root/claude:2215:5171)\n    at A (/$bunfs/root/claude:11:7259)\n    at FZ8 (/$bunfs/root/claude:4862:4342)","timestamp":"2026-01-25T12:35:42.934Z"},{"error":"Error: ENOENT: no such file or directory, scandir '/Users/lucabartelsen/.claude/skills'\n    at readdirSync (unknown)\n    at <anonymous> (/$bunfs/root/claude:12:1903)\n    at mW (/$bunfs/root/claude:11:35435)\n    at rea…

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