[codex] Outline ToolExecutor handler bodies (#27299)

## Why

We're now [discouraging use of
`async_trait`](https://github.com/openai/codex/pull/20242).

Removing use of `async_trait` from `ToolExecutor` yields a `codex_core`
debug test build speedup of ~78% (from 227.5s to 50.3s) on my machine.

For ease of reviewing, this is a prefactor to extract trait method
implementations to inherent methods. This will prevent changing
indentation from creating a huge diff.

## What

Outlined existing `ToolExecutor::handle` bodies into inherent async
`handle_call` methods across core and extension tool handlers.

The trait methods still use `async_trait` and now delegate to
`self.handle_call(...).await`; handler behavior is unchanged.

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

@@ -97,6 +97,12 @@ impl ToolExecutor<ToolCall> for ImageGenerationTool {
 
     /// Executes the selected image operation and returns the completed image result.
     async fn handle(&self, call: ToolCall) -> Result<Box<dyn ToolOutput>, FunctionCallError> {
+        self.handle_call(call).await
+    }
+}
+
+impl ImageGenerationTool {
+    async fn handle_call(&self, call: ToolCall) -> Result<Box<dyn ToolOutput>, FunctionCallError> {
         let args = parse_args(&call)?;
         let request = request_for_args(&args, call.conversation_history.items())?;
         call.turn_item_emitter

codex-rs/ext/memories/src/tools/ad_hoc_note.rs

@@ -61,6 +61,19 @@ where
         &self,
         call: ToolCall,
     ) -> Result<Box<dyn codex_extension_api::ToolOutput>, codex_extension_api::FunctionCallError>
+    {
+        self.handle_call(call).await
+    }
+}
+
+impl<B> AddAdHocNoteTool<B>
+where
+    B: MemoriesBackend,
+{
+    async fn handle_call(
+        &self,
+        call: ToolCall,
+    ) -> Result<Box<dyn codex_extension_api::ToolOutput>, codex_extension_api::FunctionCallError>
     {
         let backend = self.backend.clone();
         let args: AddAdHocNoteArgs = parse_args(&call)?;

codex-rs/ext/memories/src/tools/list.rs

@@ -59,6 +59,19 @@ where
         &self,
         call: ToolCall,
     ) -> Result<Box<dyn codex_extension_api::ToolOutput>, codex_extension_api::FunctionCallError>
+    {
+        self.handle_call(call).await
+    }
+}
+
+impl<B> ListTool<B>
+where
+    B: MemoriesBackend,
+{
+    async fn handle_call(
+        &self,
+        call: ToolCall,
+    ) -> Result<Box<dyn codex_extension_api::ToolOutput>, codex_extension_api::FunctionCallError>
     {
         let backend = self.backend.clone();
         let args: ListArgs = parse_args(&call)?;

codex-rs/ext/memories/src/tools/read.rs

@@ -58,6 +58,19 @@ where
         &self,
         call: ToolCall,
     ) -> Result<Box<dyn codex_extension_api::ToolOutput>, codex_extension_api::FunctionCallError>
+    {
+        self.handle_call(call).await
+    }
+}
+
+impl<B> ReadTool<B>
+where
+    B: MemoriesBackend,
+{
+    async fn handle_call(
+        &self,
+        call: ToolCall,
+    ) -> Result<Box<dyn codex_extension_api::ToolOutput>, codex_extension_api::FunctionCallError>
     {
         let backend = self.backend.clone();
         let args: ReadArgs = parse_args(&call)?;

codex-rs/ext/memories/src/tools/search.rs

@@ -67,6 +67,19 @@ where
         &self,
         call: ToolCall,
     ) -> Result<Box<dyn codex_extension_api::ToolOutput>, codex_extension_api::FunctionCallError>
+    {
+        self.handle_call(call).await
+    }
+}
+
+impl<B> SearchTool<B>
+where
+    B: MemoriesBackend,
+{
+    async fn handle_call(
+        &self,
+        call: ToolCall,
+    ) -> Result<Box<dyn codex_extension_api::ToolOutput>, codex_extension_api::FunctionCallError>
     {
         let backend = self.backend.clone();
         let args: SearchArgs = parse_args(&call)?;

codex-rs/ext/web-search/src/tool.rs

@@ -75,6 +75,12 @@ impl ToolExecutor<ToolCall> for WebSearchTool {
     }
 
     async fn handle(&self, call: ToolCall) -> Result<Box<dyn ToolOutput>, FunctionCallError> {
+        self.handle_call(call).await
+    }
+}
+
+impl WebSearchTool {
+    async fn handle_call(&self, call: ToolCall) -> Result<Box<dyn ToolOutput>, FunctionCallError> {
         let commands = parse_commands(&call)?;
         let command_action = command_action(&commands);
         let provider = self