I know how key pairs are used.
They’re used as I described.
Look I don't know if you got confused but I deal with key pairs as a tools programer on a daily basis. I know how they work very well. I have written a lot of software that automates aspects of their use, and even some of that recently.
The .ssh/authorize_keys files on servers are full of PUBLIC keys. Only the .ssh/id_rsa or similar files on the clients will have the PRIVATE keys.
The server requires somebody trying to use a service to prove who they are by providing a PRIVATE key which is the solution to the riddle implicit in the PUBLIC key.
PRIVATE keys are PRIVATE. They are like the password. They are NEVER stored on a server unless the server is run by hackers that phish it out of someone. They are called PRIVATE, because ONLY you...or at least the .ssh/id_rsa or some other such protected file on your personal computer or other client..are supposed to know them. And any files that have them at least on linux/unix/mac systems must have limited permissions so they are only accesible to the user (such as chmod 600). The open source ssh client software enforces this!
Private keys are kind of like passwords, but have an advantage and a disadvantage. The disadvantage is they are too damn long to remember and to type. The advantage is that they do not need to be stored on the server. The server just needs the public key to tell if the private key is right. And yet even if the public key is obtained by a hacker they can't figure out the private key from it. However if the hacker got the passwords from the server...then they could impersonate the individual. So hackers are forced to trying to phish individuals into giving their private keys away, and there is no good way for them to hack a central database and get everyone's at once.