The document you are referring to is for RDS, and not for Aurora Serverless, hence you will not be able to move the subnets and reboot since those steps are not applicable here. If you are using Aurora Serverless v1, it should already be private, even if the subnets are public. Reference: "You can't give an Aurora Serverless v1 DB cluster a public IP address. You can access an Aurora Serverless v1 DB cluster only from within a VPC."
Another option is to create a new subnet group only with private subnets, and restore a snapshot of your cluster into that subnet group.
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