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Migrates some directories off of deprecated base::Bind, base::Callback,
etc, and onto the newer APIs.

Specifically this covers files in components/search_provider_logos/

Fixed: 1007728
Change-Id: Icc664ea3b2fdfc690d0cfe32a879b4d8c071707e
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/1972240Reviewed-by: Marc Treib <treib@chromium.org>
Commit-Queue: Ken Rockot <rockot@google.com>
Cr-Commit-Position: refs/heads/master@{#726010}