FYI, I briefly looked into this flaky failure and here is what happens (I think):

1) On ChromeOS Shadow DOM v0 is used by default (same as --disable-features=WebUIPolymer2)
2) PolymerTest.testIronIcons() is a check that runs after every Polymer test, see [1]. It relies on /deep/ which only works for Shadow DOM v0.
3) There is a tracking bug about figuring out what to do with testIronIcons() and Shadow DOM v1 (delete testIronIcons(), write a recursive querySelector() helper that does not rely on /deep/ ?).

+dbeam: Who faces a similar failure.

[1] https://cs.chromium.org/chromium/src/chrome/test/data/webui/polymer_browser_test_base.js?l=112

I am able to repro the failure locally with
./out/gchromeos/browser_tests --gtest_filter=CrComponentsManagedFootnoteTest.All --disable-features=WebUIPolymer2

Looking into whether there is a simple fix.

Uploaded candidate fix at https://chromium-review.googlesource.com/c/chromium/src/+/1371083. 

The following revision refers to this bug:
  https://chromium.googlesource.com/chromium/src.git/+/be6bec8d74c853d83e9afea2d192d9d1fa463264

commit be6bec8d74c853d83e9afea2d192d9d1fa463264
Author: dpapad <dpapad@chromium.org>
Date: Tue Dec 11 02:18:54 2018

WebUI: Fix flaky <managed-footnote> tests.

The test flakes because of  PolymerTest.testIronIcons() which is problematic
in multiple ways:

1) It only works with Shadom DOM v0 (tracked by https://crbug.com/885401)
2) It suffers from timing issues, where the iron-icon being checked has not
   been fully populated yet (for example missing the internal <svg> element
   by the time the check occurs).

Clearing the DOM in teardown() works around the problem by effectively making
testIronIcons() a no-op for this test.

Bug:  913708 
Change-Id: I643f2cf457e7bd2f3f4056452f7ccce308556972
Reviewed-on: https://chromium-review.googlesource.com/c/1371083
Reviewed-by: Dan Beam <dbeam@chromium.org>
Commit-Queue: Demetrios Papadopoulos <dpapad@chromium.org>
Cr-Commit-Position: refs/heads/master@{#615386}
[modify] https://crrev.com/be6bec8d74c853d83e9afea2d192d9d1fa463264/chrome/test/data/webui/cr_components/managed_footnote_test.js