Refactor auth providers to mutate request headers (#17866)

## Summary
- Move auth header construction into the
`AuthProvider::add_auth_headers` contract.
- Inline `CoreAuthProvider` header mutation in its provider impl and
remove the shared header-map helper.
- Update HTTP, websocket, file upload, sideband websocket, and test auth
callsites to use the provider method.
- Add direct coverage for `CoreAuthProvider` auth header mutation.

## Testing
- `just fmt`
- `cargo test -p codex-api`
- `cargo test -p codex-core
client::tests::auth_request_telemetry_context_tracks_attached_auth_and_retry_phase`
- `cargo test -p codex-core` failed on unrelated/reproducible
`tools::handlers::multi_agents::tests::multi_agent_v2_followup_task_interrupts_busy_child_without_losing_message`

---------

Co-authored-by: Celia Chen <celia@openai.com>

codex-rs/codex-api/tests/sse_end_to_end.rs

@@ -53,9 +53,7 @@ impl HttpTransport for FixtureSseTransport {
 struct NoAuth;
 
 impl AuthProvider for NoAuth {
-    fn bearer_token(&self) -> Option<String> {
-        None
-    }
+    fn add_auth_headers(&self, _headers: &mut HeaderMap) {}
 }
 
 fn provider(name: &str) -> Provider {