[EKS] [request]: proxy kubectl commands through AWS to allow remote operations on private EKS endpoints

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**Tell us about your request**
Currently accessing a private endpoint can be done in 3 ways: "Connected Network", "EC2 Bastion", and "Cloud9".
I'd like to be able to proxy commands through AWS so that network access isn't necessary.
I'm imagining something similar to AWS SSM's "start-session", where SSH is proxied through AWS and no ports need to be opened.
Thanks!

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

**Tell us about the problem you're trying to solve. What are you trying to do, and why is it hard?**
Using only private endpoints is a security requirement for us.
The current options for accessing a private endpoint are all cumbersome.

**Are you currently working around this issue?**
We're using a VPN

**Additional context**
N/A

**Attachments**
N/A

The request names EKS, private endpoints, kubectl, and AWS SSM start-session as the relevant context. Start by clarifying the supported proxy flow and identifying the AWS, EKS, and kubectl entry points involved. Done would mean remote kubectl operations work against private EKS endpoints without requiring network access.

Written by the indexing model from the issue text.