[Reland] Some linked_ptr removal in chromeos file_browser_handler.

Original CL: https://chromium.googlesource.com/chromium/src/+/37ab3be841440d4ec5abf1160acf32c905796d13
was reverted due to unit_tests failure on chromeos debug bots.
There was an ownership issue in volume_manager.cc which this
CL fixes.

BUG=556939, 696375
Test=No visible changes. Ran VolumeManagerTest.* in debug, which passes now.

Review-Url: https://codereview.chromium.org/2723753002
Cr-Commit-Position: refs/heads/master@{#454742}
6 files changed