Ubuntu – Server still prompts for password after SSH configuration, Ubuntu 17.10 x64

EDIT: After consulting verbose logs as was suggested, I found the error myself. Answer is at the bottom.

I have a DigitalOcean droplet I want to use as a development environment. I've created a non-root user with sudo privileges to do all the actual work, and have set this user up with the public key in my pair via ssh-copy-id. Attempting to copy the key over again confirms that the key is present and recognized:

/usr/bin/ssh-copy-id: WARNING: All keys were skipped because they already exist on the remote system.

This is after trying the following suggestions:

• Empty the contents of .ssh on both machines, generate and copy a new key
• Use chmod to ensure that .ssh is set to 700 and authorized_keys is set to 600 on the VPS.
• Uncommented the line in etc/ssh/sshd_config stating PubkeyAuthentication yes as well as
AuthorizedKeysFile .ssh/authorized_keys .ssh/authorized_keys2
followed by restarting the ssh daemon on the VPS.
• Rebuilding the droplet and trying again from scratch
• Manually copying over the key using cat
• Trying to setup the key with the root user to then swap over to the non-root user.

At this stage, any attempt to connect via SSH still prompts for the user's password even though I explicitly set the key to be passwordless, and failing to provide this password results in this error: Permission denied (publickey,password). If I provide the password, I'm still able to log in directly as the user.

The following are the verbose logs before the password prompt appears:

OpenSSH_7.2p2 Ubuntu-4ubuntu2.2, OpenSSL 1.0.2g  1 Mar 2016
debug1: Reading configuration data /etc/ssh/ssh_config
debug1: /etc/ssh/ssh_config line 19: Applying options for *
debug2: resolving "138.197.31.224" port 22
debug2: ssh_connect_direct: needpriv 0
debug1: Connecting to 138.197.31.224 [138.197.31.224] port 22.
debug1: Connection established.
debug1: key_load_public: No such file or directory
debug1: identity file /home/user/.ssh/id_rsa type -1
debug1: key_load_public: No such file or directory
debug1: identity file /home/user/.ssh/id_rsa-cert type -1
debug1: key_load_public: No such file or directory
debug1: identity file /home/user/.ssh/id_dsa type -1
debug1: key_load_public: No such file or directory
debug1: identity file /home/user/.ssh/id_dsa-cert type -1
debug1: key_load_public: No such file or directory
debug1: identity file /home/user/.ssh/id_ecdsa type -1
debug1: key_load_public: No such file or directory
debug1: identity file /home/user/.ssh/id_ecdsa-cert type -1
debug1: key_load_public: No such file or directory
debug1: identity file /home/user/.ssh/id_ed25519 type -1
debug1: key_load_public: No such file or directory
debug1: identity file /home/user/.ssh/id_ed25519-cert type -1
debug1: Enabling compatibility mode for protocol 2.0
debug1: Local version string SSH-2.0-OpenSSH_7.2p2 Ubuntu-4ubuntu2.2
debug1: Remote protocol version 2.0, remote software version OpenSSH_7.5p1 Ubuntu-10
debug1: match: OpenSSH_7.5p1 Ubuntu-10 pat OpenSSH* compat 0x04000000
debug2: fd 3 setting O_NONBLOCK
debug1: Authenticating to 138.197.31.224:22 as 'user'
debug1: SSH2_MSG_KEXINIT sent
debug1: SSH2_MSG_KEXINIT received
debug2: local client KEXINIT proposal
debug2: KEX algorithms: curve25519-sha256@libssh.org,ecdh-sha2-nistp256,ecdh-sha2-nistp384,ecdh-sha2-nistp521,diffie-hellman-group-exchange-sha256,diffie-hellman-group-exchange-sha1,diffie-hellman-group14-sha1,ext-info-c
debug2: host key algorithms: ecdsa-sha2-nistp256-cert-v01@openssh.com,ecdsa-sha2-nistp384-cert-v01@openssh.com,ecdsa-sha2-nistp521-cert-v01@openssh.com,ecdsa-sha2-nistp256,ecdsa-sha2-nistp384,ecdsa-sha2-nistp521,ssh-ed25519-cert-v01@openssh.com,ssh-rsa-cert-v01@openssh.com,ssh-ed25519,rsa-sha2-512,rsa-sha2-256,ssh-rsa
debug2: ciphers ctos: chacha20-poly1305@openssh.com,aes128-ctr,aes192-ctr,aes256-ctr,aes128-gcm@openssh.com,aes256-gcm@openssh.com,aes128-cbc,aes192-cbc,aes256-cbc,3des-cbc
debug2: ciphers stoc: chacha20-poly1305@openssh.com,aes128-ctr,aes192-ctr,aes256-ctr,aes128-gcm@openssh.com,aes256-gcm@openssh.com,aes128-cbc,aes192-cbc,aes256-cbc,3des-cbc
debug2: MACs ctos: umac-64-etm@openssh.com,umac-128-etm@openssh.com,hmac-sha2-256-etm@openssh.com,hmac-sha2-512-etm@openssh.com,hmac-sha1-etm@openssh.com,umac-64@openssh.com,umac-128@openssh.com,hmac-sha2-256,hmac-sha2-512,hmac-sha1
debug2: MACs stoc: umac-64-etm@openssh.com,umac-128-etm@openssh.com,hmac-sha2-256-etm@openssh.com,hmac-sha2-512-etm@openssh.com,hmac-sha1-etm@openssh.com,umac-64@openssh.com,umac-128@openssh.com,hmac-sha2-256,hmac-sha2-512,hmac-sha1
debug2: compression ctos: none,zlib@openssh.com,zlib
debug2: compression stoc: none,zlib@openssh.com,zlib
debug2: languages ctos:
debug2: languages stoc:
debug2: first_kex_follows 0
debug2: reserved 0
debug2: peer server KEXINIT proposal
debug2: KEX algorithms: curve25519-sha256,curve25519-sha256@libssh.org,ecdh-sha2-nistp256,ecdh-sha2-nistp384,ecdh-sha2-nistp521,diffie-hellman-group-exchange-sha256,diffie-hellman-group16-sha512,diffie-hellman-group18-sha512,diffie-hellman-group14-sha256,diffie-hellman-group14-sha1
debug2: host key algorithms: ssh-rsa,rsa-sha2-512,rsa-sha2-256,ecdsa-sha2-nistp256,ssh-ed25519
debug2: ciphers ctos: chacha20-poly1305@openssh.com,aes128-ctr,aes192-ctr,aes256-ctr,aes128-gcm@openssh.com,aes256-gcm@openssh.com
debug2: ciphers stoc: chacha20-poly1305@openssh.com,aes128-ctr,aes192-ctr,aes256-ctr,aes128-gcm@openssh.com,aes256-gcm@openssh.com
debug2: MACs ctos: umac-64-etm@openssh.com,umac-128-etm@openssh.com,hmac-sha2-256-etm@openssh.com,hmac-sha2-512-etm@openssh.com,hmac-sha1-etm@openssh.com,umac-64@openssh.com,umac-128@openssh.com,hmac-sha2-256,hmac-sha2-512,hmac-sha1
debug2: MACs stoc: umac-64-etm@openssh.com,umac-128-etm@openssh.com,hmac-sha2-256-etm@openssh.com,hmac-sha2-512-etm@openssh.com,hmac-sha1-etm@openssh.com,umac-64@openssh.com,umac-128@openssh.com,hmac-sha2-256,hmac-sha2-512,hmac-sha1
debug2: compression ctos: none,zlib@openssh.com
debug2: compression stoc: none,zlib@openssh.com
debug2: languages ctos:
debug2: languages stoc:
debug2: first_kex_follows 0
debug2: reserved 0
debug1: kex: algorithm: curve25519-sha256@libssh.org
debug1: kex: host key algorithm: ecdsa-sha2-nistp256
debug1: kex: server->client cipher: chacha20-poly1305@openssh.com MAC: <implicit> compression: none
debug1: kex: client->server cipher: chacha20-poly1305@openssh.com MAC: <implicit> compression: none
debug1: expecting SSH2_MSG_KEX_ECDH_REPLY
debug1: Server host key: ecdsa-sha2-nistp256 SHA256:cAw/cF/p37TL25mwRKivxJpMCPyrCMFo5YaeTYQ2Wvo
debug1: Host '138.197.31.224' is known and matches the ECDSA host key.
debug1: Found key in /home/user/.ssh/known_hosts:1
debug2: set_newkeys: mode 1
debug1: rekey after 134217728 blocks
debug1: SSH2_MSG_NEWKEYS sent
debug1: expecting SSH2_MSG_NEWKEYS
debug2: set_newkeys: mode 0
debug1: rekey after 134217728 blocks
debug1: SSH2_MSG_NEWKEYS received
debug2: key: /home/user/.ssh/id_rsa ((nil))
debug2: key: /home/user/.ssh/id_dsa ((nil))
debug2: key: /home/user/.ssh/id_ecdsa ((nil))
debug2: key: /home/user/.ssh/id_ed25519 ((nil))
debug1: SSH2_MSG_EXT_INFO received
debug1: kex_input_ext_info: server-sig-algs=<ssh-ed25519,ssh-rsa,rsa-sha2-256,rsa-sha2-512,ssh-dss,ecdsa-sha2-nistp256,ecdsa-sha2-nistp384,ecdsa-sha2-nistp521>
debug2: service_accept: ssh-userauth
debug1: SSH2_MSG_SERVICE_ACCEPT received
debug1: Authentications that can continue: publickey,password
debug1: Next authentication method: publickey
debug1: Trying private key: /home/user/.ssh/id_rsa
debug1: Trying private key: /home/user/.ssh/id_dsa
debug1: Trying private key: /home/user/.ssh/id_ecdsa
debug1: Trying private key: /home/user/.ssh/id_ed25519
debug2: we did not send a packet, disable method
debug1: Next authentication method: password

If we look at the above logs, these are the most relevant lines:

debug1: Next authentication method: publickey
debug1: Trying private key: /home/user/.ssh/id_rsa
debug1: Trying private key: /home/user/.ssh/id_dsa
debug1: Trying private key: /home/user/.ssh/id_ecdsa
debug1: Trying private key: /home/user/.ssh/id_ed25519
debug2: we did not send a packet, disable method
debug1: Next authentication method: password

We can see that ssh has four default files it will check for, which is fine assuming you let ssh-keygen provide a default name during generation. However, if you give your key a special name, you will have to point ssh to the location of the private key, as it does not know to look for it. After I noticed this, I attempted to log in with the following command:

ssh user@xxx.xxx.xx.xxx -i .ssh/custom_named_rsa

. . . and was successful. Note that the above just happens to be the relative path to the key from the directory I was in, the usual considerations of pathing apply. Thanks to both suggestions to process logs, especially the one for verbose logging on the client side.