Agent forwarding is not the agent. Forwarding asks sshd on the remote host to create its own socket and point SSH_AUTH_SOCK at it, so programs on that host can ask your agent back home to sign. The remote never holds the key, and it never needs to: it holds full use of it. ...
An ssh-agent is a program that holds private keys for public key authentication and answers signing requests on their behalf. ssh finds it through SSH_AUTH_SOCK, the path of a Unix-domain socket, so the key is unlocked once and then borrowed by every connection that follows. The private key itself is never handed to a client. ...
Record the expected fingerprint, use a deliberate StrictHostKeyChecking policy, and treat changes as an incident until explained.
SSH keys are convenient, and convenience is exactly why they get out of hand. The practical discipline is not to generate fewer keys; it is to make each key a named identity with a clear scope, a short lifetime, and a record in the files that tell an auditor who is allowed to do what. ...