It sounds like you have taken the appropriate steps to grant yourself access to the built-in SQL Serverless pool in your Synapse workspace, but are still encountering an "Unauthorized" status.
One additional step you may want to try is to check if the Azure Synapse Analytics service is in the same subscription as your Azure AD tenant. If they are not in the same subscription, you may need to grant your Azure AD tenant access to the subscription containing the Synapse workspace.
To do this, navigate to the subscription that contains your Synapse workspace and select "Access control (IAM)" from the left-hand menu. From there, you can add a new role assignment and select "Azure AD user, group, or service principal" as the assignee. Then, enter the email address associated with your Azure AD tenant and select the appropriate role (e.g. "Contributor" or "Owner").
If you have already tried this step or it does not resolve the issue, you may need to contact a Synapse Administrator for your workspace to troubleshoot the issue further. They may be able to identify any underlying issues or misconfigurations that are preventing you from accessing the SQL Serverless pool.