[Desktop] Browser pane tools unavailable in SSH sessions (works in Local)

Summary
In the Claude Desktop Code tab, a session connected to a remote Linux host over SSH does not get the built-in Browser pane tools (mcp__Claude_Browser__*). A Local session on the same Mac does. So Claude can't open or drive a page in the Browser pane from an SSH session, even though the pane itself runs on the Mac.

Environment

• Claude Desktop on macOS, Code tab
• SSH session to a remote Linux VPS (aarch64, Oracle Linux 9)
• Claude Code version on remote: 2.1.280 (Claude Code)

Steps to reproduce

1. Start a Code tab session with an SSH environment.
2. Ask Claude to open a URL in the built-in browser.
3. Claude can't find any Claude_Browser tools (ToolSearch returns nothing), so it can only open the URL in the Mac's default browser.
4. Start a Local session and do the same thing: the Browser tools are there and it works.

Expected
SSH sessions get the Browser pane tools too, the same way connectors, plugins, and the file pane already work there. The pane runs locally in Desktop, so reaching externally routable URLs (such as a Tailscale IP) shouldn't depend on where the session runs.

Docs
https://code.claude.com/docs/en/desktop doesn't say whether the Browser pane is supported in SSH sessions. If this is intentional, please document it next to the other per-session-type notes (terminal = local only, file pane = local + SSH).