Don't loop CEF exit while shutting down if task post fails

[tt2468]

Description

Removes a while loop which loops infinitely if posting the shutdown task to the manager thread fails. In scenarios where Cef failed to initialize, this manager thread exits early, leaving a situation where tasks cannot be posted to it. In normal scenarios, posting the task is not expected to fail anyway.

This also logs a debug message just in case it becomes relevant in the future.

Motivation and Context

If the manager_thread exits early, there will be no message loop to post the task to, and a permanent deadlock would occur.

Rather than guaranteeing a deadlock, this at least lowers the odds of one.

How Has This Been Tested?

• Ubuntu 24.04 - X11 on NVIDIA
• Tested a standard working configuration where browsers load and then OBS is closed gracefully
• Tested a non-working configuration where CEF fails to initialize due to an invalid SingletonCookie

All test scenarios exited gracefully without deadlocking.

Types of changes

• Bug fix (non-breaking change which fixes an issue)

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.