Password Reset not sending

Hi all,
This installation uses wpsmtp mail plugin.

Plugin seems to be configured correctly: I can send a test mail and receive it correctly.
I also see the send in the log.

SMTP was never configured on the plesk server.

I even created a custom reset password function in my theme to personalized the message, it does work correctly, message received, seen in log. Receive the reset link correctly.

But I cannot truely reset the password, since the lost-password form does not send the new password.

So my question is does the wp mail need to be configured on the server ? Or should I this plugin override the default method ? Did anyone had the same problem?