Use Brave sync chain instead of Brave Account (uplift to 1.13.x)

[brave-builds]

Uplift of #6317
Resolves brave/brave-browser#10469

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]

Win x64 failed due to binaries not being uploaded into S3 as per https://ci.brave.com/job/pr-brave-browser-pr6317_fix_brave_account_strings_1.13.x-windows/1/execution/node/461/log/ and https://ci.brave.com/job/pr-brave-browser-pr6317_fix_brave_account_strings_1.13.x-windows/1/execution/node/488/log/. However, this is a known issue and has already been resolved.

15:08:48  upload failed: .\ to s3://****/brave-browser-build-pr/pr6317_fix_brave_account_strings_1.13.x/\ufffd\ufffd9 [Errno 2] No such file or directory: 
15:08:48  'C:\\13.x\\src\\out\\Release\\'
15:08:48  
15:08:49  upload failed: dist\ to s3://****/brave-browser-build-pr/pr6317_fix_brave_account_strings_1.13.x/\ufffd\ufffd9 [Errno 2] No such file or directory: 
15:08:49  'C:\\13.x\\src\\out\\Release\\dist\\'

15:08:50  Searching s3://****/brave-browser-build-pr/pr6317_fix_brave_account_strings_1.13.x/��9a3802011303cf85c29c1cfb01bb718b8902f7fb-��f35c0cd326821a80ceef0ce650bb7a78d004f7d0 for glob:'**' 

[kjozwiak]

Uplift into 1.13.x approved after deliberating with @brave/uplift-approvers. QA has also verified the PR on Nightly as per #6317 (comment).