[K8S Dogfooding] Deploy more runners and integrate more jobs from the gitlab-runner repo

We have a runner for the integration tests - Dogfooding the Kubernetes executor - Step 2 - R... (#38305 - closed) • Georgi N. Georgiev | GitLab • 17.7

We have a runner running the chart jobs - [K8S Dogfooding] Deploy a Runner to run helm ch... (#38453 - closed) • Georgi N. Georgiev | GitLab • 17.9

While the former as very specific and the latter is very simplistic, a good next target group would be the jobs of gitlab-runner itself, e.g. unit tests or binary building.