Separate ssh keys for the user and the admin? Yeah, see point 2, admins should not be remotely accessible.
Really? How, exactly? Break the ssh key authentication? And wouldn’t that apply to all accounts equally?
Keys aren't perfect security. They can easily be mishandled, sometimes getting published to GitHub, copied to USB drives which can easily be lost, etc.
Further, there have been attacks against SSH that let malicious actors connect remotely to any session, or take over existing sessions. By not allowing remote access on privileged accounts, you minimize risk.
Forcing a non privileged remote session to authenticate with a password establishes a second factor of security that is different from the first. This means a cracked password or a lost key is still not enough for a malicious actor to accomplish administrative privileges.
A key is something you have
A password is something you know
So, by not allowing remote privileged sessions, we're forcing a malicious actor to take one more non-trivial step before arriving at their goals. A step that will likely be fairly obvious in logs on a monitored machine.