If you use a different strong random password for every single server, this attack isn't a problem, but that adds a lot of management hassle compared to using a single private key. (It's also made more difficult by host key checking, but let's be honest, most of us don't diligently check the fingerprints every single time we get a mismatch warning.)
In contrast, if you use an SSH key, then a compromised server never actually gets a copy of your private key unless you explicitly copy it over. (If you're have SSH agent forwarding turned on, then during the compromised connection, the server can run a "confused deputy" attack to authenticate other connections using your agent's identity. But it loses that ability when you disconnect.)