Fixes storage not persist after reset (uplift to 1.13.x)

[brave-builds]

Uplift of #6285
Uplift of #6274
Uplift of #6284
Resolves brave/brave-browser#11024
Resolves brave/brave-browser#10934
Resolves brave/brave-browser#11035

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]

Currently PR/Blocked due to not being able to QA #6284 until #6248 has been fixed on Nightly.

[LaurenWags]

Per discussion with @NejcZdovc going to test using PR build (https://bravesoftware.slack.com/archives/C5DT3GPCH/p1596464740328400) instead of waiting for Nightly to be fixed.

[LaurenWags]

Linux failed due to known issue with RewardsBrowserTest.NotVerifiedWallet (https://ci.brave.com/job/pr-brave-browser-pr6285_reset-reducer_1.13.x/3/execution/node/508/log/), this does not block uplift.

Windows failed (https://ci.brave.com/job/pr-brave-browser-pr6285_reset-reducer_1.13.x/3/execution/node/755/log/) due to agent being removed and is being restarted.

[LaurenWags]

Restarting on Windows again, agent removed again. https://ci.brave.com/job/pr-brave-browser-pr6285_reset-reducer_1.13.x/4/execution/node/343/log/

[kjozwiak]

Restarting Win x64 as we lost the agent/instance as per Agent was removed via https://ci.brave.com/job/pr-brave-browser-pr6285_reset-reducer_1.13.x/5/execution/node/375/log/.

18:24:51  Cannot contact i-073ab23ddfa6baad0: hudson.remoting.RequestAbortedException: java.io.IOException: Unexpected termination of the channel
18:29:54  Could not connect to i-073ab23ddfa6baad0 to send interrupt signal to process

[kjozwiak]

Uplift into 1.13.x approved after deliberating with @brave/uplift-approvers. QA also verified the PRs on Nightly as per #6285 (comment), #6274 (comment) and #6284 (comment)