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This relands http://crrev.com/771122 with compile fixes.

It's safe for the Widget to own the underlying View via
SetContentsView().  This was not immediately obvious; much of this CL is
adjacent refactoring and cleanup that simplifies the ownership/lifetime.

Bug: 1044687
Change-Id: I49820e70df1387cfde6998de6ab67e119b5a451a
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/2213227
Commit-Queue: Peter Kasting <pkasting@chromium.org>
Auto-Submit: Peter Kasting <pkasting@chromium.org>
Reviewed-by: Katie Dektar <katie@chromium.org>
Cr-Commit-Position: refs/heads/master@{#772107}