This removes the title entirely to get rid of a duplicate sentence, resolving at the same time a <h2> problem... And offers a way to make the checkout choice of the date optionnal !

woocommerce_local_pickup_time_plugin_custom_2.php

<?php
/**
 * Remove the WooCommerce Local Pickup Time plugin checkout page select field title.
 * And offers a way to make the checkout choice of the date optionnal !
 *
 * @see https://github.com/WC-Local-Pickup/woocommerce-local-pickup-time/issues/103
 * @see https://github.com/WC-Local-Pickup/woocommerce-local-pickup-time/issues/104
 */
if ( class_exists( 'Local_Pickup_Time' ) ) {
    // Add a checkbox field to make the pickup date choice optionnal
    add_filter( 'woocommerce_general_settings', 'make_pickup_time_choice_mandatory', 20 );
    // Remove the default mandatory check on checkout.
    remove_action( 'woocommerce_checkout_process', array( Local_Pickup_Time::get_instance(), 'field_process') );
    // And add this custom one.
    add_action( 'woocommerce_checkout_process', 'my_not_mandatory_custom_field_process' );
    // Remove the default local pickup time field from the checkout page.
    remove_action( apply_filters( 'local_pickup_time_select_location', 'woocommerce_after_order_notes' ), array( Local_Pickup_Time::get_instance(), 'time_select') );
    // And add this custom one.
    add_action( apply_filters( 'local_pickup_time_select_location', 'woocommerce_after_order_notes' ), 'my_not_mandatory_custom_time_select' );
    // Hide the time select field if an other delivery method is choosen on checkout page
    add_action( 'wp_footer', 'hide_local_pickup_time_select' );

    /**
     * Add a checkbox field to make the pickup date choice optionnal
     *
     * @param array $settings General settings fields.
     */
    function make_pickup_time_choice_mandatory( $settings ){
        $updated_settings = array();
        
        foreach ( $settings as $setting ) {
            $updated_settings[] = $setting;
            
            if ( isset( $setting['id'] ) && 'local_pickup_days_ahead' === $setting['id'] ){
                $updated_settings[] = array(
                    'title' => 'Choix d\'une date obligatoire ?',
                    'desc'  => 'Si vous ne proposez que le retrait en magasin, vous devriez cocher cette case. Sinon, pensez bien à prévenir vos clients qui choisissent le retrait en boutique qu\'ils doivent penser à indiquer un créneau pour passer chercher leur commande.',
                    'id'    => 'local_pickup_pickup_date_mandatory',
                    'type'  => 'checkbox',
                );
            }
        }
        
        return $updated_settings;
    }

    /**
     * Process the checkout
     */
    function my_not_mandatory_custom_field_process() {
        global $woocommerce;

        // Check if mandatory
        $required = get_option( 'local_pickup_pickup_date_mandatory' );

        // Check if mandatory and set, if its mandatory and not set add an error.
        if ( 'yes' === $required && ! $_POST['local_pickup_time_select'] ) {
            wc_add_notice( __( 'Please select a pickup time.', 'woocommerce-local-pickup-time' ), 'error' );
        }
    }

    /**
     * Add the local pickup time field to the checkout page
     *
     * @param object $checkout The checkout object.
     */
    function my_not_mandatory_custom_time_select( $checkout ) {
        $required = get_option( 'local_pickup_pickup_date_mandatory' );
        echo '<div id="local-pickup-time-select">';

        $lpt = Local_Pickup_Time::get_instance();
        woocommerce_form_field(
            'local_pickup_time_select',
            array(
                'type'     => 'select',
                'class'    => array( 'local-pickup-time-select-field form-row-wide' ),
                'label'    => __( 'Pickup Time', 'woocommerce-local-pickup-time' ),
                'required' => ( 'yes' === $required ? true : false ),
                'options'  => method_exists( $lpt, 'build_pickup_time_options' ) ?  $lpt->build_pickup_time_options() : $lpt->get_pickup_time_options(),
            ),
            $checkout->get_value( 'local_pickup_time_select' )
        );

        echo '</div>';
    }

    /**
     * Hide the time select field if an other delivery method is choosen on checkout page
     *
     * @param object $checkout The checkout object.
     */
    function hide_local_pickup_time_select() {
        // Is the time select field required ?
        $required = 'yes' === get_option( 'local_pickup_pickup_date_mandatory' ) ? true : false;
        // If it's not required and we are on checkout page
        if ( is_checkout() && ! $required ) {
            ?>
            <style>
                /* Hide the menu by default */
                #local-pickup-time-select{display:none;}
            </style>
            <script type="text/javascript">
                // Wait for the page to be fully loaded
                // @see https://wordpress.stackexchange.com/questions/342148/woocommerce-documentation-list-of-js-events
                jQuery('body').on('updated_checkout',function(){
                    // Show the menu if the right delivery method is choosen
                    if(jQuery('#payment_method_cod').is(':checked')) jQuery('#local-pickup-time-select').show();
                    // Allow menu showing / hiding to take action depending of the delivery method choice
                    jQuery('#payment input[type=radio]').each(function(){
                        jQuery(this).change(function() {
                            // Show the menu if local pickup is selected
                            if ('cod' === jQuery(this).val()) jQuery('#local-pickup-time-select').show();
                            // Hide it if another method is choosen
                            else jQuery('#local-pickup-time-select').hide();
                        });
                    });
                });
            </script>
            <?php
        }
    }
}

J'ai ajouté un javascript qui cache le menu déroulant si ce n'est pas le retrait en magasin qui est choisi comme méthode de livraison. Pour que le menu reste toujours visible, il suffit de commenter la ligne 21 en plaçant devant deux slash :
// add_action( 'wp_footer', 'hide_local_pickup_time_select' );

Hello, which line of code is the one that is hiding the pop-up message "please choose the time for your pickup" that's showing (even when customer choose another method than pickup) ?

Thanks !