Migrate core test support to the shared HTTP client (#34651)

## What changed

- Send the streaming SSE request-body drain test through `HttpClientFactory`.
- Replace the test-support crate's direct `reqwest` dependency with
  `codex-http-client`.
- Remove `core_test_support` from the temporary direct-`reqwest` exceptions.

GitOrigin-RevId: 16ea4309d0b575592c2e59991eb6dce18d9467ed

codex-rs/deny.toml

@@ -250,7 +250,6 @@ deny = [
         "codex-responses-api-proxy",
         "codex-rmcp-client",
         "codex-tui",
-        "core_test_support",
         # Third-party crates that own their reqwest integration. These are not part of the
         # first-party migration count above.
         "oauth2",