[script] use dart:io support for forwarding stdout/stderr (#5410)

diff --git a/script/tool/lib/src/common/process_runner.dart b/script/tool/lib/src/common/process_runner.dart
index 8ac98b8..1be6488 100644
--- a/script/tool/lib/src/common/process_runner.dart
+++ b/script/tool/lib/src/common/process_runner.dart
@@ -36,11 +36,9 @@
     print(
         'Running command: "$executable ${args.join(' ')}" in ${workingDir?.path ?? io.Directory.current.path}');
     final io.Process process = await io.Process.start(executable, args,
-        workingDirectory: workingDir?.path, environment: environment);
-    await Future.wait(<Future<dynamic>>[
-      io.stdout.addStream(process.stdout),
-      io.stderr.addStream(process.stderr),
-    ]);
+        workingDirectory: workingDir?.path,
+        environment: environment,
+        mode: io.ProcessStartMode.inheritStdio);
     if (exitOnError && await process.exitCode != 0) {
       final String error =
           _getErrorString(executable, args, workingDir: workingDir);