Add logging to the classes related to the new chat

What does this MR do and why?

Add logging in various places in GitLab chat related code to allow for more visibility during development and after 🚀

Screenshots or screen recordings

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Before	After

How to set up and validate locally

Numbered steps to set up and validate the change are strongly suggested.

MR acceptance checklist

This checklist encourages us to confirm any changes have been analyzed to reduce risks in quality, performance, reliability, security, and maintainability.

• I have evaluated the MR acceptance checklist for this MR.

Related to #412994