Can't connect to Cisco router using network_cli but ssh from raw module works fine

Good! My My pleasure.

That’s configuration item listed in Ansible Configuration Settings.
https://docs.ansible.com/ansible/latest/reference_appendices/config.html#paramiko-look-for-keys

According to my research, by default it seams paramiko searches a default keys (such as ~/.ssh/id_rsa) even though the ansible_password is defined.
The paramiko uses ansible_password by setting look_for_keys = False

It is unclear why the error message "msg": "No existing session" appears…

Could you mark as solution my comment for explicitly state that this topic has been resolved?