Mysql – Optimization of a query on a table that represents a weak entity type

I've been trying to figure this problem out for a while now, so I decided I'll try my luck here.
I have a very complex online shop DB structure (multiple shops in one database, hundreds of thousands of products) and I'm querying MySQL to return products from a specified category. Products are in a many-to-many (M:N) relationship to categories, so I have tables product, category and product_category (which stands for a weak entity type).

Table schemas are as follows:

CREATE TABLE `product` (
  `id` int(11) NOT NULL AUTO_INCREMENT,
  `name` varchar(255) COLLATE utf8_czech_ci NOT NULL,
  `description` text COLLATE utf8_czech_ci NOT NULL,
  `date` timestamp NOT NULL DEFAULT current_timestamp(),
  `date_edited` timestamp NOT NULL DEFAULT current_timestamp(),
  `visibility` tinyint(1) NOT NULL DEFAULT 1,
  `main_slider` tinyint(1) NOT NULL DEFAULT 0,
  `bought` int(11) NOT NULL DEFAULT 0,
  `price_action` int(11) NOT NULL DEFAULT 0,
  `quantity_action` int(11) NOT NULL DEFAULT 0,
  `duration_action` timestamp NOT NULL DEFAULT current_timestamp(),
  `new` tinyint(1) NOT NULL DEFAULT 1,
  `type` int(2) NOT NULL DEFAULT 0,
  `origin_url` varchar(250) COLLATE utf8_czech_ci DEFAULT NULL,
  `origin_price` int(11) NOT NULL DEFAULT 0,
  `origin_price_dph` int(11) NOT NULL DEFAULT 0,
  `rating` int(11) NOT NULL DEFAULT 0,
  `ratingcount` int(11) NOT NULL DEFAULT 0,
  `facebook_flag` tinyint(1) DEFAULT 0,
  `salebot` tinyint(1) NOT NULL DEFAULT 0,
  `shop_supplier_id` int(11) NOT NULL,
  `shop_id` int(11) NOT NULL,
  `product_global_id` int(11) DEFAULT NULL,
  `prioritize` tinyint(1) NOT NULL DEFAULT 0,
  `free_delivery` tinyint(1) NOT NULL DEFAULT 0,
  PRIMARY KEY (`id`),
  UNIQUE KEY `product_global_id_2` (`product_global_id`,`shop_id`),
  KEY `shop_id` (`shop_id`),
  KEY `shop_supplier_id` (`shop_supplier_id`),
  KEY `product_global_id` (`product_global_id`),
  KEY `rating` (`rating`),
  KEY `date` (`date`),
  KEY `quantity_action` (`quantity_action`),
  KEY `duration_action` (`duration_action`),
  KEY `main_slider` (`main_slider`),
  KEY `shop_id_2` (`shop_id`,`main_slider`),
  KEY `shop_id_3` (`shop_id`,`prioritize`),
  KEY `shop_id_4` (`shop_id`,`date`)
) ENGINE=InnoDB DEFAULT CHARSET=utf8 COLLATE=utf8_czech_ci;

CREATE TABLE `product_category` (
  `category_id` int(11) NOT NULL,
  `product_id` int(11) NOT NULL,
  PRIMARY KEY (`category_id`,`product_id`),
  KEY `category` (`category_id`),
  KEY `product` (`product_id`)
) ENGINE=InnoDB DEFAULT CHARSET=utf8 COLLATE=utf8_czech_ci;

And the query I'm using:

SELECT product.id, 
       product.name, 
       product.shop_supplier_id, 
       product.price_action, 
       product.quantity_action, 
       product.duration_action, 
       variant.id as variant_id, 
       IF(variant.price_default IS NULL, variant_global.price_vat, variant.price_default) as price_default, 
       variant.price_dph as price, 
       variant_global.delivery_date, 
       variant.value AS variant_name, 
       IF(product.type = 0, variant_global.voc_silver, variant.voc_dph) as voc_dph, 
       shop_supplier.text_available, 
       shop_supplier.text_unavailable, 
       IF(product.type = 0, variant_global.available = 1, variant.availability = 1) as avail, 
       variant.availability_type, 
       supplier.id as supplier_id, 
       variant_global.price_vat as price_default_ds, 
       product.type 
      FROM `product` 
      LEFT JOIN shop_supplier ON (`product`.`shop_supplier_id` = `shop_supplier`.`id`) 
      LEFT JOIN supplier ON (shop_supplier.supplier_id = `supplier`.`id`)      
      LEFT JOIN variant_global ON ((SELECT id FROM variant_global WHERE variant_global.product_global_id=product.product_global_id ORDER BY `available` DESC LIMIT 1) = `variant_global`.`id`) 
      LEFT JOIN product_global ON (variant_global.product_global_id = product_global.id) 
      LEFT JOIN image ON (`product`.`id` = `image`.`product_id`) 
      LEFT JOIN variant ON ((SELECT id FROM variant WHERE `product_id`=product.id ORDER BY `availability` DESC LIMIT 1) = `variant`.`id`) 
      LEFT JOIN product_category ON (`product`.`id` = `product_category`.`product_id`) 
        WHERE (product.shop_id=100000) 
          AND (product.visibility= 1) 
          AND (image.id IS NOT NULL) 
          AND (IF(product.type = 0, variant_global.available = 1, variant.availability = 1) = (CASE WHEN display_available = 1 AND display_unavailable = 1 THEN IF(product.type = 0, variant_global.available = 1, variant.availability = 1) WHEN display_available = 1 THEN 1 ELSE 0 END)) 
          AND (variant_global.deleted IS NULL OR variant_global.deleted = 0) 
          AND (product_category.category_id= 5) 
     GROUP BY product.id 
     ORDER BY prioritize DESC, RAND() LIMIT 5;

The query is called multiple times on a single page for different categories and is really slow, I've tried all different keys and combinations and couldn't figure it out.

The EXPLAIN of the query outputs this:

+------+--------------------+------------------+--------+-----------------------------------------------+---------------------+---------+-------------------------------------+------+----------------------------------------------+
| id   | select_type        | table            | type   | possible_keys                                 | key                 | key_len | ref                                 | rows | Extra                                        |
+------+--------------------+------------------+--------+-----------------------------------------------+---------------------+---------+-------------------------------------+------+----------------------------------------------+
|    1 | PRIMARY            | product_category | ref    | PRIMARY,category,product                      | PRIMARY             | 4       | const                               | 4264 | Using index; Using temporary; Using filesort |
|    1 | PRIMARY            | product          | eq_ref | PRIMARY,shop_id,shop_id_2,shop_id_3,shop_id_4 | PRIMARY             | 4       | dropohs.product_category.product_id |    1 | Using where                                  |
|    1 | PRIMARY            | shop_supplier    | eq_ref | PRIMARY                                       | PRIMARY             | 4       | dropohs.product.shop_supplier_id    |    1 |                                              |
|    1 | PRIMARY            | supplier         | eq_ref | PRIMARY                                       | PRIMARY             | 4       | dropohs.shop_supplier.supplier_id   |    1 | Using where; Using index                     |
|    1 | PRIMARY            | variant          | eq_ref | PRIMARY                                       | PRIMARY             | 4       | func                                |    1 | Using where                                  |
|    1 | PRIMARY            | image            | ref    | PRIMARY,product_id_2,product_id               | product_id_2        | 4       | dropohs.product_category.product_id |    1 | Using where; Using index                     |
|    1 | PRIMARY            | variant_global   | eq_ref | PRIMARY                                       | PRIMARY             | 4       | func                                |    1 | Using where                                  |
|    3 | DEPENDENT SUBQUERY | variant          | ref    | product_id,product_id_2                       | product_id_2        | 4       | dropohs.product.id                  |    1 | Using where; Using index                     |
|    2 | DEPENDENT SUBQUERY | variant_global   | ref    | product_global_id,product_global_id_2         | product_global_id_2 | 4       | dropohs.product.product_global_id   |    1 | Using where; Using index                     |
+------+--------------------+------------------+--------+-----------------------------------------------+---------------------+---------+-------------------------------------+------+----------------------------------------------+

If you need any further information or schemas of the other tables used in the query, let me know, but I omited them for now as I don't think they are the cause of the problem as all of them are properly using indexes. Thanks.