Add checkbox to product inventory tab in WooCommerce and have the checkbox checked by default

woocommerce_admin_process_product_object replaces the outdated woocommerce_process_product_meta hook so you should definitely not replace it with it

To have the checkbox checked by default you can add value to the args from woocommerce_wp_checkbox()

So you get:

// Add checkbox
function action_woocommerce_product_options_inventory_product_data() {
    global $product_object;
    
    // Get meta
    $value = $product_object->get_meta( '_cutom_meta_key' );
    
    // Checkbox
    woocommerce_wp_checkbox( array( 
        'id'            => '_cutom_meta_key', // Required, it's the meta_key for storing the value (is checked or not)
        'label'         => __( 'Custom label', 'woocommerce' ), // Text in the editor label
        'desc_tip'      => false, // true or false, show description directly or as tooltip
        'description'   => __( 'Enable this to make something', 'woocommerce' ), // Provide something useful here
        'value'         => empty( $value ) ? 'yes' : $value // Checked by default
    ) );
}
add_action( 'woocommerce_product_options_inventory_product_data', 'action_woocommerce_product_options_inventory_product_data', 10, 0 );
        
// Save Field
function action_woocommerce_admin_process_product_object( $product ) {
    // Update meta
    $product->update_meta_data( '_cutom_meta_key', isset( $_POST['_cutom_meta_key'] ) ? 'yes' : 'no' );
}
add_action( 'woocommerce_admin_process_product_object', 'action_woocommerce_admin_process_product_object', 10, 1 );