ReplicationIT: Remove unnecessary storage inspection

Integration tests shouldn't need to rely on inspecting the underlying
ReplicationTasksStorage layer(s). All of these tests already verify the
expected end result.

This leaves 4 tests that currently completely rely on inspecting the
task storage to verify the expected result. Those tests need further
improvement to decouple from the storage layer.

Change-Id: I029d63ce7d07414d9bf5d9290d556378beedcabf