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This patch migrates tests away from deprecated APIs and adopts
TestingProfile::Builder for tests in
/chrome/browser/engagement.

Rationale: creating or overriding keyed services after the profile has
been created is problematic and known to cause hard-to-debug test
flakiness, because it bypasses BrowserContextDependencyManager and often
leading to use-after-free.

This CL was uploaded by git cl split.

R=calamity@chromium.org

Bug: 1106699
Change-Id: Ia256324372fffe91b7dd7bda603faea66fbbb666
Reviewed-on: https://chromium-review.googlesource.com/c/chromium/src/+/2316361
Auto-Submit: Mikel Astiz <mastiz@chromium.org>
Reviewed-by: calamity <calamity@chromium.org>
Commit-Queue: calamity <calamity@chromium.org>
Cr-Commit-Position: refs/heads/master@{#792498}