Move Incoming Email checks to System Checks

What does this MR do?

It migrates the SystemCheck from the check.rake to the SystemCheck format.

Are there points in the code the reviewer needs to double check?

No

Why was this MR needed?

This is part of the effort of #28080 (moved)

Does this MR meet the acceptance criteria?

• Changelog entry added, if necessary
• Documentation created/updated
• API support added
• Tests added for this feature/bug
• Review
  • Has been reviewed by UX
  • Has been reviewed by Frontend
  • Has been reviewed by Backend
  • Has been reviewed by Database
• Conform by the merge request performance guides
• Conform by the style guides
• Squashed related commits together

What are the relevant issue numbers?

This is part of #28080 (moved)