Fixed default SR topsites are disappeared when toggling themes option (uplift to 1.13.x)

[brave-builds]

Uplift of #6261
Resolves brave/brave-browser#11009

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• 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]

Looks like Android timed out as per https://ci.brave.com/job/pr-brave-browser-pr6261_fix_topsites_super_referrals_1.13.x/1/execution/node/218/log/.

23:19:21  Cannot contact ecs-android-android-ecs-g3sdb: hudson.remoting.ChannelClosedException: Channel "hudson.remoting.Channel@124fdb74:JNLP4-connect connection from ip-172-31-42-166.us-west-2.compute.internal/172.31.42.166:38412": Remote call on JNLP4-connect connection from ip-172-31-42-166.us-west-2.compute.internal/172.31.42.166:38412 failed. The channel is closing down or has closed down
01:19:05  Could not connect to ecs-android-android-ecs-g3sdb to send interrupt signal to process

[kjozwiak]

@simonhong this one might need a rebase as we've merged #6276 & brave/brave-browser#11026 into 1.12.x and 1.13.x.

[simonhong]

@kjozwiak rebased and pushed.

[kjozwiak]

CI looks good as continuous-integration/jenkins/pr-head appears green 👍 This PR was created/passed before @mihaiplesa implemented the separate pipelines for each platform.

[kjozwiak]

Uplift into 1.13.x approved after deliberating with @brave/uplift-approvers. QA has verified the PR on Nightly as per #6261 (comment). @bsclifton also verified via #6261 (review).