See also: https://github.com/FiloSottile/whoami.filippo.io
Using a private key per system or set of systems means that such an attacker would only gain access to what that key grants them access to.
My "github" key only grants access to my github repos, not to my bitbucket repos or to my DigitalOcean servers, or my Scaleway servers.
I have an SSH key for each "thing", and each key is kept in a different keyring via https://github.com/ccontavalli/ssh-ident, _precisely_ to avoid that problem.
As an example, an attacker gaining root on my Oracle Cloud free server will not be able to clone all my Github repos with it: I only use that key to connect to that server, and while I may have the github key unlocked at the same time - that's on a different keyring which the server has no visibility of.