Do not run web shard during tool_tests when subshard is missing (#87385)

diff --git a/dev/bots/test.dart b/dev/bots/test.dart
index 3469eb9..2c8348d 100644
--- a/dev/bots/test.dart
+++ b/dev/bots/test.dart
@@ -140,7 +140,8 @@
       'framework_coverage': _runFrameworkCoverage,
       'framework_tests': _runFrameworkTests,
       'tool_tests': _runToolTests,
-      'web_tool_tests': _runToolTests,
+      // web_tool_tests is also used by HHH: https://dart.googlesource.com/recipes/+/refs/heads/master/recipes/dart/flutter_engine.py
+      'web_tool_tests': _runWebToolTests,
       'tool_integration_tests': _runIntegrationToolTests,
       // All the unit/widget tests run using `flutter test --platform=chrome`
       'web_tests': _runWebUnitTests,
@@ -355,7 +356,6 @@
   await selectSubshard(<String, ShardRunner>{
     'general': _runGeneralToolTests,
     'commands': _runCommandsToolTests,
-    'web': _runWebToolTests,
   });
 }