Deprecate content::BrowserThread::PostAfterStartupTask

BEST_EFFORT tasks will not run until after startup and only if there is no
critical work pending.

This patch should not change the current behaviour. Please double check
whether the task being posted via BrowserThread::PostAfterStartupTask
really cares about being run after startup or just wants to run after
any critical work was completed. If you are fine with it just running
after critical work has completed LGTM and keep on with your life. If
that is not the case let me know and please explain your reasoning in a
comment.

More details:
https://docs.google.com/document/d/1P8PV1qX2cOR2WRFD6v9CjygJkE_mftCtJ7BoqQHCKZI

This CL was uploaded by git cl split.

R=raymes@chromium.org

Bug: 887407
Change-Id: Ia9dc2635bcaf61cfec8134f381260e9d19d70752
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/1632240
Commit-Queue: calamity <calamity@chromium.org>
Reviewed-by: calamity <calamity@chromium.org>
Auto-Submit: Carlos Caballero <carlscab@google.com>
Cr-Commit-Position: refs/heads/master@{#666197}

chrome/browser/engagement/site_engagement_service.cc

@@ -188,10 +188,8 @@ SiteEngagementService::GetAllDetailsInBackground(
 
 SiteEngagementService::SiteEngagementService(Profile* profile)
     : SiteEngagementService(profile, base::DefaultClock::GetInstance()) {
-  content::BrowserThread::PostAfterStartupTask(
-      FROM_HERE,
-      base::CreateSingleThreadTaskRunnerWithTraits(
-          {content::BrowserThread::UI}),
+  base::PostTaskWithTraits(
+      FROM_HERE, {content::BrowserThread::UI, base::TaskPriority::BEST_EFFORT},
       base::BindOnce(&SiteEngagementService::AfterStartupTask,
                      weak_factory_.GetWeakPtr()));