chore: rm web-search-eligible header (#10660)

default-enablement of web_search is now client-side, no need to send
eligibility headers to backend.

Tested locally, headers no longer sent.

will wait for corresponding backend change to deploy before merging

codex-rs/core/src/compact.rs

@@ -20,7 +20,6 @@ use crate::truncate::TruncationPolicy;
 use crate::truncate::approx_token_count;
 use crate::truncate::truncate_text;
 use crate::util::backoff;
-use codex_protocol::config_types::WebSearchMode;
 use codex_protocol::items::ContextCompactionItem;
 use codex_protocol::items::TurnItem;
 use codex_protocol::models::ContentItem;
@@ -352,10 +351,6 @@ async fn drain_to_completed(
     turn_metadata_header: Option<&str>,
     prompt: &Prompt,
 ) -> CodexResult<()> {
-    let web_search_eligible = !matches!(
-        turn_context.config.web_search_mode,
-        Some(WebSearchMode::Disabled)
-    );
     let mut stream = client_session
         .stream(
             prompt,
@@ -363,7 +358,6 @@ async fn drain_to_completed(
             &turn_context.otel_manager,
             turn_context.reasoning_effort,
             turn_context.reasoning_summary,
-            web_search_eligible,
             turn_metadata_header,
         )
         .await?;