Browser tests are run from
  ChromeBrowserMainParts::PreMainMessageLoopRun
    ChromeBrowserMainParts::PreMainMessageLoopeRunImpl
      parameters().ui_task->Run() in chrome_browser_main.cc:1827

which is before ChromeBrowserMainParts::PreMainMessageLoopRun calls
PreMainMessageLoopRun on extra parts. Since our services were being
started in extra parts PreMainMessageLoopRun, the test would run prior
to services starting.
ChromeBrowserMainParts::PostBroserStart, however, is being called prior
to execution of ui_task from which the tests are run.