Turn bridge into `running` before creating downstream

What does this MR do and why?

This is a change for this issue #356759 (comment 1109775441).

Also checkout the feature flag issue #377789 (closed)

In order to display and calculate the duration for a pipeline that has :depend bridge jobs, the started_at needs to be set.

How to set up and validate locally

1. Create a project
2. Add .gitlab-ci.yml with:

run-child-pipeline:   # This job runs in the test stage.
  trigger:
    include: ci/child-pipeline.yml
    strategy: depend

3. Add ci/child-pipeline.yml with:

child-job:       # This job runs in the build stage, which runs first.
  script:
    - echo "Running child job..."
    - sleep 20
    - echo "Running complete."

4. Commit changes and trigger the CI build
5. You should see the duration appears in the pipeline listing page for this build.

MR acceptance checklist

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