Remove codex-core's direct reqwest dependency (#36076)

## Why

`codex-http-client` is the intended owner of `reqwest`, while direct first-party
dependencies are tracked as migration debt.

## What changed

- Use `http` header and status types directly in core code and tests.
- Build the MCP test readiness client through `HttpClientBuilder`.
- Remove `reqwest` from `codex-core` and its temporary dependency-policy
  exception.

GitOrigin-RevId: 8d5c7c86256294a7bb8aa1ae10cc5bea4cb1fc56

codex-rs/deny.toml

@@ -243,7 +243,6 @@ deny = [
         "codex-responses-api-proxy",
         # Temporary migration exceptions.
         "codex-app-server",
-        "codex-core",
         "codex-ollama",
         "codex-otel",
         # Third-party crates that own their reqwest integration. These are not part of the