UI: Refactor system tray

[cg2121]

Description

This moves the system tray to its own file.

Motivation and Context

Instead of being strewn all over OBSBasic, the system tray code is now consolidated into its own file, making it easier to maintain.

How Has This Been Tested?

Used system tray to perform actions in OBS.

Types of changes

• Code cleanup (non-breaking change which makes code smaller or more readable)

Checklist:

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

[tytan652]

The frontend API should not be consumed in the UI, it should be done alike to #7278.

[cg2121]

#7278 will need to be merged first in order to use signals instead of the frontend api. So until that happens, I'll mark this as draft for now.