Use Gitlab::Git's Popen on that module's code

What does this MR do?

Gitaly Conversation: gitaly#540 (closed)

Use Gitlab::Git's Popen on that module's code

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

No

Why was this MR needed?

This allows the current Gitaly migration to depend on less code outside of the Gitlab::Git module

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?

Adjustment necessary for gitaly!344 (comment 40062298)

Closes gitaly#597 (closed)