feat(mcp): per-server startup timeout (#3182)

Seeing timeouts on certain, slow mcp server starting up when codex is
invoked. Before this change, the timeout was a hard-coded 10s. Need the
ability to define arbitrary timeouts on a per-server basis.

## Summary of changes

- Add startup_timeout_ms to McpServerConfig with 10s default when unset
- Use per-server timeout for initialize and tools/list
- Introduce ManagedClient to store client and timeout; rename
LIST_TOOLS_TIMEOUT to DEFAULT_STARTUP_TIMEOUT
- Update docs to document startup_timeout_ms with example and options
table

---------

Co-authored-by: Matthew Dolan <dolan-openai@users.noreply.github.com>

docs/config.md

@@ -334,6 +334,8 @@ Defines the list of MCP servers that Codex can consult for tool use. Currently,
 
 **Note:** Codex may cache the list of tools and resources from an MCP server so that Codex can include this information in context at startup without spawning all the servers. This is designed to save resources by loading MCP servers lazily.
 
+Each server may set `startup_timeout_ms` to adjust how long Codex waits for it to start and respond to a tools listing. The default is `10_000` (10 seconds).
+
 This config option is comparable to how Claude and Cursor define `mcpServers` in their respective JSON config files, though because Codex uses TOML for its config language, the format is slightly different. For example, the following config in JSON:
 
 ```json
@@ -358,6 +360,8 @@ Should be represented as follows in `~/.codex/config.toml`:
 command = "npx"
 args = ["-y", "mcp-server"]
 env = { "API_KEY" = "value" }
+# Optional: override the default 10s startup timeout
+startup_timeout_ms = 20_000
 ```
 
 ## shell_environment_policy
@@ -574,6 +578,7 @@ Options that are specific to the TUI.
 | `mcp_servers.<id>.command` | string | MCP server launcher command. |
 | `mcp_servers.<id>.args` | array<string> | MCP server args. |
 | `mcp_servers.<id>.env` | map<string,string> | MCP server env vars. |
+| `mcp_servers.<id>.startup_timeout_ms` | number | Startup timeout in milliseconds (default: 10_000). Timeout is applied both for initializing MCP server and initially listing tools. |
 | `model_providers.<id>.name` | string | Display name. |
 | `model_providers.<id>.base_url` | string | API base URL. |
 | `model_providers.<id>.env_key` | string | Env var for API key. |