[MCP] Remove the legacy stdio client in favor of rmcp (#5529)

I haven't heard of any issues with the studio rmcp client so let's
remove the legacy one and default to the new one.

Any code changes are moving code from the adapter inline but there
should be no meaningful functionality changes.

codex-rs/core/src/features.rs

@@ -31,7 +31,7 @@ pub enum Feature {
     UnifiedExec,
     /// Use the streamable exec-command/write-stdin tool pair.
     StreamableShell,
-    /// Use the official Rust MCP client (rmcp).
+    /// Enable experimental RMCP features such as OAuth login.
     RmcpClient,
     /// Include the freeform apply_patch tool.
     ApplyPatchFreeform,