## 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
Michael Bolin
89aeae6c21