Follow up of #6547 - Brave.Shields.CookieListEnabled response is not sent to local state file

[GeetaSarvadnya]

Description:

Follow up of #6547 - Brave.Shields.CookieListEnabled response is not sent to local state file

Steps to Reproduce

1. Clean profile Brave Dev 1.46 (22.12.6.11)
2. Open Youtube.com
3. Remove app from memory and restart the browser
4. Cookies consent prompt is shown
5. Click "Block cookie notices" on the prompt
6. Open local state file via settings
7. Brave.Shields.CookieListEnabled response is 0 instead of 1

Note: Removed app from memory and relaunched, value is not changed to 1

Actual result:
Brave.Shields.CookieListEnabled response is 0 instead of 1

Expected result:
Brave.Shields.CookieListEnabled response should be 1 when "Block cookie notices" prompt is clicked

Reproduces how often: [Easily reproduced, Intermittent Issue]
Easy

Brave Version:
Brave Dev 1.46 (22.12.6.11)

Device details:
iPhone 8 - 16.1.1

Website problems only:

• did you check with Brave Shields down?NA
• did you check in Safari/Firefox (WkWebView-based browsers)? NA

cc: @brave/qa-team @kylehickinson @anthonypkeane

Additional Information

[GeetaSarvadnya]

Verification PASSED on iPhone 8 version 16.1.1 running Brave Dev 1.46 (22.12.6.19)

• Verified the cases from the description and ensured that the Brave.Shields.CookieListEnabled response is 1 when "Block cookie notices" prompt is clicked