[EKS] [Fargate]: Support to Fluent Bit for EKS Fargate in a private cluster

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**Tell us about your request**
What do you want us to build?
Configure Fluent Bit for EKS Fargate in a private cluster

**Which service(s) is this request for?**
Fargate, EKS

**Tell us about the problem you're trying to solve. What are you trying to do, and why is it hard?**
I have tried to configure Fluent Bit for EKS Fargate[1] in a private cluster[2]. Despite the required VPC endpoints exist and accessible by the pods and also the events of the new pods show that logging is enabled, the log group is not created.

It only seems to work when a NAT Gateway (internet connection) is associated with the route tables of the subnets,

**Are you currently working around this issue?**
Sidecar containers seems to be the valid workaround for the timebeing

Reference:
[1] https://aws.amazon.com/blogs/containers/fluent-bit-for-amazon-eks-on-aws-fargate-is-here/
[2] https://docs.aws.amazon.com/eks/latest/userguide/private-clusters.html

No repository files or tests are named. Start with the linked Fluent Bit for EKS Fargate and private-cluster references, then investigate why logging works with a NAT Gateway but not the listed VPC endpoints; done means Fluent Bit creates the log group and delivers logs in a private cluster without internet access.

Written by the indexing model from the issue text.