Extract CI config processor into pipeline chain

What does this MR do?

This is the 3rd attempt to move the config processor out of Ci::Pipeline. This time we are splitting it into several steps of the Pipeline::Chain (idea from @ayufan)

First attempt: !19562 (closed) Second attempt: !19818 (closed)

The 3rd attempt combines the idea of splitting the config processor and all its related methods into different steps of the Pipeline::Chain as well as other refactoring steps and spec changes from the first attempt.

Related issue: #35727 (closed)

Screenshots

Does this MR meet the acceptance criteria?

Conformity

• Changelog entry
• Documentation (if required)
• Code review guidelines
• Merge request performance guidelines
• Style guides
• Database guides
• Separation of EE specific content

Availability and Testing

• Review and add/update tests for this feature/bug. Consider all test levels. See the Test Planning Process.
• Tested in all supported browsers

Security

If this MR contains changes to processing or storing of credentials or tokens, authorization and authentication methods and other items described in the security review guidelines:

• Label as security and @ mention @gitlab-com/gl-security/appsec
• The MR includes necessary changes to maintain consistency between UI, API, email, or other methods
• Security reports checked/validated by a reviewer from the AppSec team