We found out why it did not work:
login-with-ajax is incompatible with the WP mail SMTP plug-in wp-config.php option:
Dashboard -> WP mail SMTP
tells:
SMTP Password
[ …… ]
The password is stored in plain text. We highly recommend you set up your password in your WordPress configuration file for improved security.
To do this add the lines below to your wp-config.php file:
define( ‘WPMS_ON’, true );
define( ‘WPMS_SMTP_PASS’, ‘your_password’ );
After removing those 2 lines from the wp-config.php file login-with-ajax started to send the emails.