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The current approach isn't maintainable, and a new param is needed for
better coverage of incognito and guest mode profiles.

Bug: b/166730281
Change-Id: Iea8b7ec6ced3867d83fc5224499830db9c6f53bb
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/2397982
Commit-Queue: Trent Apted <tapted@chromium.org>
Reviewed-by: Giovanni Ortuño Urquidi <ortuno@chromium.org>
Reviewed-by: Jiewei Qian  <qjw@chromium.org>
Reviewed-by: Xiyuan Xia <xiyuan@chromium.org>
Cr-Commit-Position: refs/heads/master@{#805616}