Can't make Ansible to wait for a server to reboot and continue playbook to work

This is a commonly known problem. See Reboot a server and wait for it to come back. Since Ansible 1.9.4 SSH loses connection before proceeding to the next task.

You need to add a delay (sleep) before the shutdown command:

- name: restart server
  shell: sleep 2 && shutdown -r now
  async: 1
  poll: 0
  become: yes
  become_method: sudo
  ignore_errors: true
- name: waiting for server to come back after reboot
  wait_for_connection:
....

On Ansible 2.7, the reboot module was introduced, which solves this problem.

You can use:

- name: restart server
  reboot: