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This moves the logic for getting a list of devices from SharingService
to a new class. The SharedClipboardMessageHandler can then use the new
class instead of the dependency on SharingService.

Bug: 1021980
Change-Id: Ie1b0146b6aaf60949481ac161bfd69905cf7c803
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/1910097
Commit-Queue: Richard Knoll <knollr@chromium.org>
Reviewed-by: Alex Chau <alexchau@chromium.org>
Cr-Commit-Position: refs/heads/master@{#715289}