chore: update exec crate to use std::time instead of chrono (#952)

When I originally wrote `elapsed.rs`, I realized we were using both
`std::time` and `chrono` with no real benefit of having both. We should
try to keep the `exec` subcommand trim (as it also buildable as a
standalone executable), so this helps tighten things up.

codex-rs/exec/src/event_processor.rs

@@ -17,6 +17,7 @@ use owo_colors::OwoColorize;
 use owo_colors::Style;
 use shlex::try_join;
 use std::collections::HashMap;
+use std::time::Instant;
 
 /// This should be configurable. When used in CI, users may not want to impose
 /// a limit so they can see the full transcript.
@@ -76,7 +77,7 @@ impl EventProcessor {
 
 struct ExecCommandBegin {
     command: Vec<String>,
-    start_time: chrono::DateTime<Utc>,
+    start_time: Instant,
 }
 
 /// Metadata captured when an `McpToolCallBegin` event is received.
@@ -84,11 +85,11 @@ struct McpToolCallBegin {
     /// Formatted invocation string, e.g. `server.tool({"city":"sf"})`.
     invocation: String,
     /// Timestamp when the call started so we can compute duration later.
-    start_time: chrono::DateTime<Utc>,
+    start_time: Instant,
 }
 
 struct PatchApplyBegin {
-    start_time: chrono::DateTime<Utc>,
+    start_time: Instant,
     auto_approved: bool,
 }
 
@@ -133,7 +134,7 @@ impl EventProcessor {
                     call_id.clone(),
                     ExecCommandBegin {
                         command: command.clone(),
-                        start_time: Utc::now(),
+                        start_time: Instant::now(),
                     },
                 );
                 ts_println!(
@@ -208,7 +209,7 @@ impl EventProcessor {
                     call_id.clone(),
                     McpToolCallBegin {
                         invocation: invocation.clone(),
-                        start_time: Utc::now(),
+                        start_time: Instant::now(),
                     },
                 );
 
@@ -263,7 +264,7 @@ impl EventProcessor {
                 self.call_id_to_patch.insert(
                     call_id.clone(),
                     PatchApplyBegin {
-                        start_time: Utc::now(),
+                        start_time: Instant::now(),
                         auto_approved,
                     },
                 );