core: route realtime and memories through HTTP client factory

[bolinfest]

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.

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Stack created with Sapling. Best reviewed with ReviewStack.

• codex-api: route file uploads through HTTP client factory #31363
• -> core: route realtime and memories through HTTP client factory #31362
• model-provider: route model discovery through HTTP client factory #31361
• core: use HTTP Responses with system proxy #31342