fix(mcp): give stdio servers the environment, and let builtins reconnect

[MummIndia]

Summary

Two defects that combine to make the built-in browser MCP server unusable, each of
which hides the other.

_connect_stdio built env={**os.environ, **env} if env else None. None does not
mean "inherit the parent environment" — the MCP SDK substitutes a minimal default one.
Callers that pass an env inherit everything, which is why the Python builtins work:
they pass builtin_python_env(base_dir). The NPX browser server passes nothing, so it
loses the entire container environment including PLAYWRIGHT_BROWSERS_PATH. It then
looks for browsers in the default cache and reports Browser "firefox" is not installed — with the browser sitting one directory away, which is what makes this
hard to place.

Second, a stdio session can disappear without the process dying: the teardown races
across asyncio tasks. call_tool returned early on a missing session, and the existing
recovery only ran when a call raised — which presupposes a session. A missing one was
therefore terminal, even though reconnecting would have fixed it. Reconnection is now
attempted in that case too.

_reconnect_builtin also excluded the browser outright: it tested membership against
_BUILTIN_SERVERS, the Python-server dict, while is_builtin() counts the NPX servers
as builtins as well. It now handles both kinds.

Target branch

• This PR targets dev, not main.

Linked Issue

Fixes #5769

Type of Change

• Bug fix (non-breaking — fixes a confirmed issue)

Checklist

• I searched open issues and open PRs — this is not a duplicate.
• This PR targets dev
• My changes are limited to the scope described above — no unrelated refactors or whitespace changes mixed in.
• I actually ran the app (docker compose up) and verified the change works end-to-end.

How to Test

1. Run in a deployment where PLAYWRIGHT_BROWSERS_PATH points somewhere other than the
   default cache, with the built-in browser server enabled.
2. Before — the server starts but reports Browser "firefox" is not installed.
   Confirm the env is the cause by printing os.environ inside the spawned server.
3. After — the server connects and reports its 30 tools, and
   browser_navigate against a real URL returns exit_code=0.
4. For the reconnect path: drop the session (kill the npx child), then call a browser
   tool. Before, every subsequent call returns
   MCP server not connected: builtin_browser for the process lifetime. After, the
   log shows No session for builtin builtin_browser; attempting reconnect followed by
   Reconnected builtin MCP server: Built-in: Browser, and the call succeeds.

Visual / UI changes

None. src/mcp_manager.py only — transport and lifecycle, no rendering path.