This is a common pattern, typically called a “jump host” or “bastion host”.
a script to cat my priv key through the relay
When it comes to security, I typically recommend against rolling your own. SSH already has agent forwarding option to do this securely and the -J option to accomplish the same without even needing to forward the key. The agent can seem complex at first, it’s actually pretty simple and worth learning.
Feel free to message me if you have more questions, I’ve got lots of experience w/ SSH.