Hello Benjamin NOAILLY - TEAMWORK,
Thank you for connecting via the email and sharing the resolution. Since the Microsoft Q&A community has a policy that "The question author cannot accept their own answer. They can only accept answers by others ", I'll repost your solution in case you'd like to "Accept " the answer.
Issue: AZURE SYNAPSE ANALYTICS SQL POOL SERVERLESS is unavailable
Solution:
The issue was related to the local machine, and it was resolved after the helpdesk addressed the local machine issue.
If I missed anything please let me know and I'd be happy to add it to my answer, or feel free to comment below with any additional information.
Please remember to "Accept Answer" if any answer/reply helped, so that others in the community facing similar issues can easily find the solution.