[Follow up to #8579] Screen time is not triggered for a website that opened in Private tab

[hffvld]

Description:

Follow-up to #8579

Steps to Reproduce

1. Device settings > Screen time > App limits
2. Set a limit for any website, for example, youtube.com or theverge.com
3. Launch Brave > Settings > Brave Shields & Privacy
4. Scroll down and turn on Enable Screen Time > Close the settings
5. Tap the view opened tabs button > Private
6. NTP > Go to youtube.com (or theverge.com) > Observe that the screen time is not triggered

Actual result:

Screen time is not triggered for a website that opened in the Private tab. The same issue was observed in Safari.

2024-01-30_09-28-41.mp4

Expected result:

Screen time must be triggered for a website that opened in the Private tab.

Reproduces how often: [Easily reproduced, Intermittent Issue]

Easily reproduced

Brave Version:

Brave build: 1.62 (24.1.29.12)
BraveCore: 1.62.153 (121.0.6167.85)

• Can you reproduce this issue with the most recent build from TestFlight? YES
• Can you reproduce this issue with the previous version of the current build from TestFlight? N/A
• Can you reproduce this issue with the current build from AppStore? N/A

Device details:

iPhone 14 / iOS 17.4 Beta

Website problems only:

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

Additional Information