After Further research, I found an open issue on gitlab pretty similar to mine.
They describe runners freshly created and unable to pick up jobs.
As for my use case, The runner was created but with warning stating "New Runner, has not connected yet".
In order to further check the status of the runner, I went to my terminal on the machine where I installed and registered the runner, namely my mac book pro, and I ran the following command:
sudo gitlab-runner verify
In the settings of the project, the runner's status switches to "Online" as you can see below, with a green label:
As a result, the project's pipeline status went from "Pending" to "Running"
Nevertheless, while the change of pipeline status, from "pending" to "running" thanks to gitlab runner's status going "Online", the associated CI build Job Status is still stuck at "Pending". This means that gitlab runner, even with all its characteristics looking fine, fails to pick up the job he's supposed to.
I'm aware this is not really a pure answer with final solution but I hope this will help move things forward on this topic !