Fix #11444: Google accounts should not show in the profile menu (uplift to 1.14.x)

[jumde]

Uplift of #6516
Resolves brave/brave-browser#11444

Approved, please ensure that before merging:

• You have checked CI and the builds, lint, and tests all pass or are not related to your PR.
• You have tested your change on Nightly.
• The PR milestones match the branch they are landing to.

After you merge:

• The associated issue milestone is set to the smallest version that the changes is landed on.

[kjozwiak]

Restarting macOS via https://ci.brave.com/job/brave-core-build-pr-macos/job/PR-6552/2/ as it seems like CI timed out during the build phase and never completed as per https://ci.brave.com/job/pr-brave-browser-profile_manager_1.14.x-macos/1/execution/node/174/log/.

17:43:34  [23905/43062] CXX obj/mojo/public/mojom/base/base_blink/memory_allocator_dump_cross_process_uid.mojom-blink.o
21:13:39  Could not connect to mac-mini2-ky to send interrupt signal to process

Restarting Win x64 via https://ci.brave.com/job/brave-core-build-pr-windows/job/PR-6552/2/ as test-install failed as per https://ci.brave.com/job/pr-brave-browser-profile_manager_1.14.x-windows/1/execution/node/240/log/. Usually we merge even though test-install failed but because macOS was restarted, might as well restart Win x64 as they'll be completed around the same time.

[kjozwiak]

test-browser failed on macOS as per https://ci.brave.com/job/pr-brave-browser-profile_manager_1.14.x-macos/3/execution/node/221/log/. However, these are known intermittent failures and don't block uplift.

00:07:48  2 tests failed:
00:07:48      RewardsContributionBrowserTest.SplitProcessOneTimeTip (../../brave/components/brave_rewards/browser/test/rewards_contribution_browsertest.cc:859)
00:07:48      RewardsContributionBrowserTest.SplitProcessorAutoContribution 

[LaurenWags]

Uplift to 1.14.x has been approved after deliberating with @brave/uplift-approvers. QA has verified this PR on Nightly as per #6516 (comment).