https://chromium-review.googlesource.com/c/chromium/src/+/1344880 seems the culprit.

Reverting CL (https://crbug.com/920040) seems to have stopped the failures, removing from sheriff queue.

The following revision refers to this bug:
  https://chromium.googlesource.com/chromium/src.git/+/009fb5f11796d3db3acda99fcf72e3b7892c304c

commit 009fb5f11796d3db3acda99fcf72e3b7892c304c
Author: Joshua Pawlicki <waffles@chromium.org>
Date: Fri Jan 18 21:44:47 2019

Reland 0a1b4fb27a194a6c4ec3fa22040a4e8152cd45b
(https://chromium-review.googlesource.com/c/chromium/src/+/1344880)

Relative to the original CL, this changes the setter for skipping
scheduled updates into a global value that can be set by browser tests
prior to constructing and kicking off the extension updater. This is
the pattern used by content verification and other extension-behavior
customization.

It's my hope that taking this approach will eliminate a lot of long-
standing flakiness in several of the tests, so if this sticks I can
follow it up with a CL re-enabling several tests. (Or another person
who has a better relationship with extensions can.)

OCL description:
> Simplify extension update check delay after browser start.
>
> Bug: 907219
> Change-Id: I067fcb333b43f60441e1789cac8caf5b996b7e5d
> Reviewed-on: https://chromium-review.googlesource.com/c/1344880
> Commit-Queue: Joshua Pawlicki <waffles@chromium.org>
> Reviewed-by: Gabriel Charette <gab@chromium.org>
> Reviewed-by: Sergey Poromov <poromov@chromium.org>
> Reviewed-by: Brian White <bcwhite@chromium.org>
> Reviewed-by: Devlin <rdevlin.cronin@chromium.org>
> Cr-Commit-Position: refs/heads/master@{#620845}

Bug: 907219, 920040
Change-Id: I35ff0311f84294eca6cd1b23a3fd612d53ea42db
Reviewed-on: https://chromium-review.googlesource.com/c/1412796
Auto-Submit: Joshua Pawlicki <waffles@chromium.org>
Reviewed-by: Devlin <rdevlin.cronin@chromium.org>
Reviewed-by: Brian White <bcwhite@chromium.org>
Reviewed-by: Sergey Poromov <poromov@chromium.org>
Reviewed-by: Gabriel Charette <gab@chromium.org>
Commit-Queue: Joshua Pawlicki <waffles@chromium.org>
Cr-Commit-Position: refs/heads/master@{#624302}
[modify] https://crrev.com/009fb5f11796d3db3acda99fcf72e3b7892c304c/chrome/browser/extensions/extension_browsertest.h
[modify] https://crrev.com/009fb5f11796d3db3acda99fcf72e3b7892c304c/chrome/browser/extensions/updater/extension_updater.cc
[modify] https://crrev.com/009fb5f11796d3db3acda99fcf72e3b7892c304c/chrome/browser/extensions/updater/extension_updater.h
[modify] https://crrev.com/009fb5f11796d3db3acda99fcf72e3b7892c304c/chrome/browser/extensions/updater/extension_updater_unittest.cc
[modify] https://crrev.com/009fb5f11796d3db3acda99fcf72e3b7892c304c/chrome/browser/policy/policy_browsertest.cc
[modify] https://crrev.com/009fb5f11796d3db3acda99fcf72e3b7892c304c/chrome/browser/prefs/browser_prefs.cc
[modify] https://crrev.com/009fb5f11796d3db3acda99fcf72e3b7892c304c/extensions/browser/extension_prefs.cc
[modify] https://crrev.com/009fb5f11796d3db3acda99fcf72e3b7892c304c/extensions/browser/pref_names.cc
[modify] https://crrev.com/009fb5f11796d3db3acda99fcf72e3b7892c304c/extensions/browser/pref_names.h
[modify] https://crrev.com/009fb5f11796d3db3acda99fcf72e3b7892c304c/tools/metrics/histograms/histograms.xml