Magento – Set all associated product not visible individually

Have a look at the Mage_Catalog_Helper_Product function canShow. This function is used by the product controller when checking to see if a product can be shown. It uses the functions isVisibleInCatalog and isVisibleInSiteVisibility

public function canShow($product, $where = 'catalog')
{
    if (is_int($product)) {
        $product = Mage::getModel('catalog/product')->load($product);
    }

    /* @var $product Mage_Catalog_Model_Product */

    if (!$product->getId()) {
        return false;
    }

    return $product->isVisibleInCatalog() && $product->isVisibleInSiteVisibility();
}

Not possible by default, no. You could try to write an observer for catalog_product_prepare_save, which is dispatched from the product save controller and receives the parameters product and request.

But with a single save, only one store view can be saved, so you cannot achieve what you want with manipulating the product object alone. You have to save the visibility attribute directly. Fortunately, there is a method for it:

addAttributeUpdate($code, $value, $store)

$product->addAttributeUpdate('visibility', false, $storeId);