How to hide username on wordpress registration?

Well I was overthinking this way too much!!

All you need to do for the form is put in a dummy value for user_login so that it looks like:

<input class="form-control" type="hidden" name="user_login" id="user_login" value="ABC">

On the top of the page I remove the $current_username = $current_email;.

On the form submission code, I make the form use $_POST['user_email'] as the user_login value. 

if (isset($_POST['user_email']))
            {

                $user_email = $_POST['user_email'];
        $user_login = $_POST['user_email'];

                // Check captch if enabled
                if (class_exists("ReallySimpleCaptcha"))
                {
                    $captcha_instance = new ReallySimpleCaptcha();

                    if ( !isset($_POST['cuar_captcha_prefix']) || !isset($_POST['captcha']))
                    {
                        $this->form_errors[] = new WP_Error('captcha',
                            __("You must write the code displayed in the image above.", 'cuarlf'));

                        return;
                    }
                    else
                    {
                        $prefix = $_POST['cuar_captcha_prefix'];
                        $code = $_POST['captcha'];

                        $captcha_checked = $captcha_instance->check($prefix, $code);
                        $captcha_instance->remove($prefix);
                        $captcha_instance->cleanup();

                        if ( !$captcha_checked)
                        {
                            $this->form_errors[] = new WP_Error('captcha', __("The code you entered is not correct.", 'cuarlf'));

                            return;
                        }
                    }
                }

                $errors = CUAR_WPLoginHelper::register_new_user($user_login, $user_email);
                if (is_wp_error($errors))
                {
                    $this->form_errors[] = $errors;

                    return;
                }
            }
            else
            {
                $this->form_errors[] = new WP_Error('user_login', __("You must enter a valid username and a valid email address.", 'cuarlf'));

                return;
            }

            $lf_addon = $this->plugin->get_addon('login-forms');
            $this->form_messages[] = sprintf(__('An email has been sent with the instructions to activate your account. '
                . 'You can then go to the <a href="https://wordpress.stackexchange.com/questions/270751/%1$s" class="alert-link">login page</a>', 'cuarlf'), $lf_addon->get_login_url());
            $this->should_print_form = false;
        }

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