go back to auto-enabling web_search for azure (#10820)

###### What
Remove special-casing that prevented auto-enabling `web_search` for
Azure model provider users. Addresses #10071, #10257.

###### Why
Azure fixed their responsesapi implementation; `web_search` is now
supported on models it wasn't before (like `gpt-5.1-codex-max`).

This request now works:
```
curl "$AZURE_API_ENDPOINT" -H "Content-Type: application/json" -H "Authorization: Bearer $AZURE_API_KEY" -d '{
  "model": "gpt-5.1-codex-max",
  "tools": [
    { "type": "web_search" }
  ],
  "tool_choice": "auto",
  "input": "Find the sunrise time in Paris today and cite the source."
}'
```

###### Tests
Tested with above curl, removed Azure-specific tests.

codex-rs/core/src/codex.rs

@@ -719,7 +719,6 @@ impl Session {
         per_turn_config.personality = session_configuration.personality;
         per_turn_config.web_search_mode = Some(resolve_web_search_mode_for_turn(
             per_turn_config.web_search_mode,
-            session_configuration.provider.is_azure_responses_endpoint(),
             session_configuration.sandbox_policy.get(),
         ));
         per_turn_config.features = config.features.clone();

codex-rs/core/src/config/mod.rs

@@ -1288,15 +1288,11 @@ fn resolve_web_search_mode(
 
 pub(crate) fn resolve_web_search_mode_for_turn(
     explicit_mode: Option<WebSearchMode>,
-    is_azure_responses_endpoint: bool,
     sandbox_policy: &SandboxPolicy,
 ) -> WebSearchMode {
     if let Some(mode) = explicit_mode {
         return mode;
     }
-    if is_azure_responses_endpoint {
-        return WebSearchMode::Disabled;
-    }
     if matches!(sandbox_policy, SandboxPolicy::DangerFullAccess) {
         WebSearchMode::Live
     } else {
@@ -2413,14 +2409,14 @@ trust_level = "trusted"
 
     #[test]
     fn web_search_mode_for_turn_defaults_to_cached_when_unset() {
-        let mode = resolve_web_search_mode_for_turn(None, false, &SandboxPolicy::ReadOnly);
+        let mode = resolve_web_search_mode_for_turn(None, &SandboxPolicy::ReadOnly);
 
         assert_eq!(mode, WebSearchMode::Cached);
     }
 
     #[test]
     fn web_search_mode_for_turn_defaults_to_live_for_danger_full_access() {
-        let mode = resolve_web_search_mode_for_turn(None, false, &SandboxPolicy::DangerFullAccess);
+        let mode = resolve_web_search_mode_for_turn(None, &SandboxPolicy::DangerFullAccess);
 
         assert_eq!(mode, WebSearchMode::Live);
     }
@@ -2429,20 +2425,12 @@ trust_level = "trusted"
     fn web_search_mode_for_turn_prefers_explicit_value() {
         let mode = resolve_web_search_mode_for_turn(
             Some(WebSearchMode::Cached),
-            false,
             &SandboxPolicy::DangerFullAccess,
         );
 
         assert_eq!(mode, WebSearchMode::Cached);
     }
 
-    #[test]
-    fn web_search_mode_for_turn_disables_for_azure_responses_endpoint() {
-        let mode = resolve_web_search_mode_for_turn(None, true, &SandboxPolicy::DangerFullAccess);
-
-        assert_eq!(mode, WebSearchMode::Disabled);
-    }
-
     #[test]
     fn profile_legacy_toggles_override_base() -> std::io::Result<()> {
         let codex_home = TempDir::new()?;

codex-rs/core/src/model_provider_info.rs

@@ -8,7 +8,6 @@
 use crate::auth::AuthMode;
 use crate::error::EnvVarError;
 use codex_api::Provider as ApiProvider;
-use codex_api::is_azure_responses_wire_base_url;
 use codex_api::provider::RetryConfig as ApiRetryConfig;
 use http::HeaderMap;
 use http::header::HeaderName;
@@ -174,10 +173,6 @@ impl ModelProviderInfo {
         })
     }
 
-    pub(crate) fn is_azure_responses_endpoint(&self) -> bool {
-        is_azure_responses_wire_base_url(&self.name, self.base_url.as_deref())
-    }
-
     /// If `env_key` is Some, returns the API key for this provider if present
     /// (and non-empty) in the environment. If `env_key` is required but
     /// cannot be found, returns an error.