Adding a public key to ~/.ssh/authorized_keys does not log me in automatically

SELinux can also cause authorized_keys not to work. Especially for root in CentOS 6 and 7. There isn't any need to disable it though.

Once you've verified your permissions are correct, you can fix this like so:

chmod 700 /root/.ssh
chmod 600 /root/.ssh/authorized_keys
restorecon -R -v /root/.ssh

You need to verify the permissions of the authorized_keys file and the folder / parent folders in which it is located.

chmod 700 ~/.ssh
chmod 600 ~/.ssh/authorized_keys

For more information see this page.

You may also need to change/verify the permissions of your home directory to remove write access for the group and others.

chmod go-w ~