UI: Use shared cookie manager for YT Control Panel

[msuman-google]

Description

Use the cookie manager shared by service integration browser docks for YouTube Control Panel.

NOTE: This commit does not have any migration logic for existing logged-in users of YTCP, they will need to sign-in again. Based on usage stats, this is not going to affect a large fraction of OBS users.

Motivation and Context

This will enable users of the YouTube Chat panel have a better (creator facing) experience for observed chat message latency, for those users who sign-in to the YouTube Control Panel or YouTube Live Chat.

How Has This Been Tested?

Built OBS locally on Linux workstation, verified existing functionality for YouTube Control Panel continues to work.

• Sign-in provides access to the control panel with stream stats and controls.
• Sign-in improves chat message latency when Chat panel is open (since it shares the cookies).
• Sign-out clears cookies from the shared cookie manager.

Types of changes

• Tweak (non-breaking change to improve existing functionality)
• Performance enhancement (non-breaking change which improves efficiency)

Checklist:

• [x] My code has been run through clang-format.
• [x] I have read the contributing document.
• [x] My code is not on the master branch.
• [x] The code has been tested.
• [x] All commit messages are properly formatted and commits squashed where appropriate.
• [x] I have included updates to all appropriate documentation.

[msuman-google]

Gentle ping for review.