Move web integration tool tests to web.shard (#70226)

* Move web integration tool tests to web.shard

Web integration tool tests depend on DDC changes in SDK. This change
moves them to a separate shard and subshard so CI bot configurations
can run them separately.

In particular, with will allow running those tests on dart CI flutter
HHH web bot instead of a non-web one, allowing early detection and easy
classification of issues caused by SDK changes as VM- or Web related.

* Enabled verbose mode for flaky web_tool_tests

* Split out the test changes to be commited first