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This splits the DRT-specific parts of ShellBrowserContext out into a new
class that we'll only instantiate when the flag is set.

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This is a re-land of https://codereview.chromium.org/637843003, which
was reverted due to layout test timeouts.
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BUG=420994

Review URL: https://codereview.chromium.org/641343002

Cr-Commit-Position: refs/heads/master@{#299092}