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This matches the behaviour of ClipboardAuraX11, which
allows OmniboxViewViewsClipboardTest to use the test clipboard.

Bug: 963112
Change-Id: I59cdc24bb7ecd2bd904505e38df133b31ed48104
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/1630012Reviewed-by: Daniel Cheng <dcheng@chromium.org>
Reviewed-by: Michael Wasserman <msw@chromium.org>
Commit-Queue: David Bienvenu <davidbienvenu@chromium.org>
Cr-Commit-Position: refs/heads/master@{#663968}