This is bassically reapplying #71721, but only enables it on web tests.

There are known issues that several tests under the `integration.shard`
depend on a specific platform, and as a result they require some
additional flexiblity (bots need to build more than one engine, and the
test flags should allow for secondary engines to be picked by such
tests).

By enabling this on the web-test shard, we will reduce the false
positives in the dart-flutter-HHH-web bot.

Fixing the more general problem is tracked by #72368.