SSH access gateway for many servers
That's too complicated (checking if a key has access to a specific prod server). Use the gateway server as jump host that accepts every valid key (but can easily remove access for a specific key which removes access to all servers in turn) and then add only the allowed keys to each respective server. After that, make sure you can reach the SSH port of every server only via the jump host.
This is the standard approach.
Engineers should not be running ansible directly from their laptop, unless this is a dev/test environment.
Instead, have a central server that pulls the runbooks from git. This allows for additional controls (four eyes, code review).
Combine this with a bastion or jump-host to restrict access further.