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We have been advised to avoid proliferating KeyedServiceFactories and
instead use one omni-service. In line with this, we move the context
manager.

Bug: b/172006764, b/170596931
Change-Id: Iebc52681839b1476c0b01f93948c48334c80162f
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/2562799Reviewed-by: Nic Hollingum <hollingum@google.com>
Reviewed-by: Daniel Ng <danielng@google.com>
Commit-Queue: Nic Hollingum <hollingum@google.com>
Cr-Commit-Position: refs/heads/master@{#833174}