Stop client from being state carrier (#10595)

I'd like to make client session wide. This requires shedding all random
state it has to carry.

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

@@ -158,8 +158,7 @@ impl Session {
         };
 
         let timer = turn_context
-            .client
-            .get_otel_manager()
+            .otel_manager
             .start_timer("codex.turn.e2e_duration_ms", &[])
             .ok();