Magento – Get Visibility of product Magento 2

Don't save a product from a collection.
A product collection item does not contain all the data for a product. You can end up losing a lot of attribute values if the attribute is not in the collection.
If you want to call $product->save() to update a product, just to be on the safe side, first you must call $product->load().

If you want to update just one attribute (or a few of them) use this. It's faster.

Mage::getModel('catalog/product_action')->updateAttributes(
    array($productId),
    array('attribute_code'=>$attributeValue),
    $storeId //use '0' for default values
);

The code above works for many products at a time, and many attributes at a time.

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();
}