Move backup results logic to service workers

[SergeyZhukovsky]

Description

Steps to Reproduce

Actual result:

Expected result:

Reproduces how often:

Desktop Brave version:

Android Device details:

• Install type (ARM, x86):
• Device type (Phone, Tablet, Phablet):
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Version/Channel Information:

• Can you reproduce this issue with the current release?
• Can you reproduce this issue with the beta channel?
• Can you reproduce this issue with the nightly channel?

Other Additional Information:

• Does the issue resolve itself when disabling Brave Shields?
• Does the issue resolve itself when disabling Brave Rewards?
• Is the issue reproducible on the latest version of Chrome?

Miscellaneous Information: