Prefer ContextClient mixin over manual ExecutionContext handling.

The ContextClient mixin interface bundles up the safe handling of
references to ExecutionContexts. Prefer using it over keeping explicit
Member<ExecutionContext> references.

R=
BUG=610176

Review-Url: https://codereview.chromium.org/2622533002
Cr-Commit-Position: refs/heads/master@{#442250}
6 files changed