[codex] Use standalone tools for Responses Lite (#26490)

## Summary

Responses Lite does not execute hosted Responses tools, so models using
it must route web search and image generation through Codex-owned
executors & standalone Response's API endpoints.

This PR is stacked on #26487.

## Validation

- `cargo test -p codex-core responses_lite_ --lib`
- `cargo test -p codex-core
standalone_executors_remain_hidden_without_flags_or_responses_lite
--lib`
- `cargo test -p codex-core
hosted_tools_follow_provider_auth_model_and_config_gates --lib`
- `cargo test -p codex-web-search-extension -p
codex-image-generation-extension`
- `cargo test -p codex-app-server --test all standalone_`
- `cargo fmt --all -- --check`

codex-rs/ext/image-generation/src/extension.rs

@@ -9,7 +9,6 @@ use codex_extension_api::ThreadStartInput;
 use codex_extension_api::ToolCall;
 use codex_extension_api::ToolContributor;
 use codex_extension_api::ToolExecutor;
-use codex_features::Feature;
 use codex_login::AuthManager;
 use codex_model_provider::create_model_provider;
 use codex_model_provider_info::ModelProviderInfo;
@@ -25,7 +24,7 @@ struct ImageGenerationExtension {
 
 #[derive(Clone)]
 struct ImageGenerationExtensionConfig {
-    enabled: bool,
+    available: bool,
     provider: ModelProviderInfo,
     codex_home: AbsolutePathBuf,
 }
@@ -34,8 +33,8 @@ impl From<&Config> for ImageGenerationExtensionConfig {
     /// Resolves whether standalone image generation should be available for a thread.
     fn from(config: &Config) -> Self {
         Self {
-            enabled: config.features.enabled(Feature::ImageGenExt)
-                && config.model_provider.is_openai(),
+            // Core selects this executor per turn using the feature flag or model metadata.
+            available: config.model_provider.is_openai(),
             provider: config.model_provider.clone(),
             codex_home: config.codex_home.clone(),
         }
@@ -75,7 +74,7 @@ impl ToolContributor for ImageGenerationExtension {
         let Some(config) = thread_store.get::<ImageGenerationExtensionConfig>() else {
             return Vec::new();
         };
-        if !config.enabled || !self.auth_manager.current_auth_uses_codex_backend() {
+        if !config.available || !self.auth_manager.current_auth_uses_codex_backend() {
             return Vec::new();
         }
 
@@ -90,7 +89,7 @@ impl ToolContributor for ImageGenerationExtension {
     }
 }
 
-/// Installs the feature-gated standalone image-generation extension contributors.
+/// Installs the standalone image-generation extension contributors.
 pub fn install(registry: &mut ExtensionRegistryBuilder<Config>, auth_manager: Arc<AuthManager>) {
     let extension = Arc::new(ImageGenerationExtension { auth_manager });
     registry.thread_lifecycle_contributor(extension.clone());