[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/Cargo.lock

@@ -1061,7 +1061,6 @@ dependencies = [
  "codex-apply-patch",
  "codex-async-utils",
  "codex-file-search",
- "codex-mcp-client",
  "codex-otel",
  "codex-protocol",
  "codex-rmcp-client",
@@ -1249,19 +1248,6 @@ dependencies = [
  "wiremock",
 ]
 
-[[package]]
-name = "codex-mcp-client"
-version = "0.0.0"
-dependencies = [
- "anyhow",
- "mcp-types",
- "serde",
- "serde_json",
- "tokio",
- "tracing",
- "tracing-subscriber",
-]
-
 [[package]]
 name = "codex-mcp-server"
 version = "0.0.0"