refactor: Move DefaultApiClient out of GitLabService

• Context MR: Draft: wip(web-ide): Use DefaultApiClient when ... (!1402 - closed)
• Part 1: refactor: Move DefaultApiClient out of GitLabSe... (!1418 - merged)
• Part 2: feat(web-ide): Use auth provider if available (!1426 - merged)

Description

This MR refactors common/.../api_client.ts out of desktop/.../gitlab_service.ts. This is helpful because the Web IDE will soon get rid of mediator requests.

Related Issues

#1253 (closed)

How has this been tested?

• No actual changes expected.
• Shouldn't effect browser/ build.
• Smoke tested by running the Run Extension task, looking at logs while opening web views and MR's.

Screenshots (if appropriate)