core: route realtime and memories through HTTP client factory (#31362)

## Why

`ModelClient` already carries the `HttpClientFactory` resolved from
session configuration, but realtime call creation and memory
summarization still constructed the legacy default client directly.
Consequently, those first-party API requests could ignore
`features.respect_system_proxy` even when Responses traffic honored it.

These are the final direct default-client constructions in
`core/src/client.rs`, so they form one small migration unit on top of
#31361.

## What changed

- Generalize `build_responses_transport` to `build_api_transport`.
- Route realtime call creation through the helper using the selected
provider and `/realtime/calls` destination.
- Route `/memories/trace_summarize` through the same helper.
- Remove the now-unused direct `build_reqwest_client` import.

## Review guide

1. The helper rename at the bottom of `core/src/client.rs` is mechanical
and keeps existing Responses behavior unchanged.
2. The realtime call path computes the route from the final provider,
including `api_provider_override`.
3. The memories path supplies its existing endpoint to the same API
route class.

## Validation

- `cargo check --tests -p codex-core`
- `just fix -p codex-core`

## Follow-up

Direct HTTP clients outside `ModelClient` remain separate migration
slices.














---
[//]: # (BEGIN SAPLING FOOTER)
Stack created with [Sapling](https://sapling-scm.com). Best reviewed
with [ReviewStack](https://reviewstack.dev/openai/codex/pull/31362).
* #31637
* #31431
* #31363
* __->__ #31362

codex-rs/core/src/client.rs

@@ -70,7 +70,6 @@ use codex_login::CodexAuth;
 use codex_login::RefreshTokenError;
 use codex_login::UnauthorizedRecovery;
 use codex_login::default_client::build_default_reqwest_client_for_route;
-use codex_login::default_client::build_reqwest_client;
 use codex_otel::SessionTelemetry;
 use codex_otel::current_span_w3c_trace_context;
 use codex_protocol::auth::AuthMode;
@@ -155,6 +154,7 @@ const WS_REQUEST_HEADER_RESPONSES_LITE_CLIENT_METADATA_KEY: &str =
 const RESPONSES_WEBSOCKETS_V2_BETA_HEADER_VALUE: &str = "responses_websockets=2026-02-06";
 const X_OPENAI_INTERNAL_CODEX_RESPONSES_LITE_HEADER: &str =
     "x-openai-internal-codex-responses-lite";
+const REALTIME_CALLS_ENDPOINT: &str = "/realtime/calls";
 const RESPONSES_ENDPOINT: &str = "/responses";
 const RESPONSES_COMPACT_ENDPOINT: &str = "/responses/compact";
 // `/responses/compact` is unary, so the timeout covers the full response rather than one idle
@@ -549,7 +549,7 @@ impl ModelClient {
         }
         let client_setup = self.current_client_setup().await?;
         let transport =
-            self.build_responses_transport(&client_setup.api_provider, RESPONSES_COMPACT_ENDPOINT)?;
+            self.build_api_transport(&client_setup.api_provider, RESPONSES_COMPACT_ENDPOINT)?;
         let request_telemetry = Self::build_request_telemetry(
             session_telemetry,
             AuthRequestTelemetryContext::new(
@@ -651,8 +651,8 @@ impl ModelClient {
         sideband_headers.extend(sideband_websocket_auth_headers(
             client_setup.api_auth.as_ref(),
         ));
-        let transport = ReqwestTransport::new(build_reqwest_client());
         let api_provider = api_provider_override.unwrap_or(client_setup.api_provider);
+        let transport = self.build_api_transport(&api_provider, REALTIME_CALLS_ENDPOINT)?;
         let response = ApiRealtimeCallClient::new(transport, api_provider, client_setup.api_auth)
             .create_with_session_and_headers(sdp, session_config, extra_headers)
             .await
@@ -682,7 +682,8 @@ impl ModelClient {
         }
 
         let client_setup = self.current_client_setup().await?;
-        let transport = ReqwestTransport::new(build_reqwest_client());
+        let transport =
+            self.build_api_transport(&client_setup.api_provider, MEMORIES_SUMMARIZE_ENDPOINT)?;
         let request_telemetry = Self::build_request_telemetry(
             session_telemetry,
             AuthRequestTelemetryContext::new(
@@ -961,7 +962,7 @@ impl ModelClient {
         })
     }
 
-    fn build_responses_transport(
+    fn build_api_transport(
         &self,
         api_provider: &ApiProvider,
         endpoint: &str,
@@ -1413,7 +1414,7 @@ impl ModelClientSession {
             let client_setup = self.client.current_client_setup().await?;
             let transport = self
                 .client
-                .build_responses_transport(&client_setup.api_provider, RESPONSES_ENDPOINT)?;
+                .build_api_transport(&client_setup.api_provider, RESPONSES_ENDPOINT)?;
             let request_auth_context = AuthRequestTelemetryContext::new(
                 client_setup.auth.as_ref().map(CodexAuth::auth_mode),
                 client_setup.api_auth.as_ref(),