Generate & secure private keys in a private network

I have seen latest mobile authentication solution where private key is generated and kept securely in a Secure Enclave/TrustZone and public key is sent to authentication server on a cloud. During User authentication, the private key is used to sign a message which can only be verified by the paired public key held by the authentication server.

Based on above concept, is it a good idea to generate cryptographic keys for each user and save private key in a secure private network somewhere in a directory or database ? and use that to verify the identity of a user on an authentication server which is on cloud?

Additional context: User details are in a private network and auth server is on cloud.