The optimization process and recommendations:

1. Create Optimal Indexes (modified query below): The recommended indexes are an integral part of this optimization effort and should be created before testing the execution duration of the optimized query.
2. Replace Left Join With Subquery (modified query below): The pattern of inflating the amount of data (using joins) and deflating (using GROUP BY) usually slows down queries. In this case, it can be avoided by moving some of the logic to the SELECT clause, and therefore removing some of the LEFT JOINs. In some cases, this transformation can lead to an obsolete GROUP BY clause, which can also be removed.

Optimal indexes for this query:

ALTER TABLE `buyout_calculator_query` ADD INDEX `buyout_query_idx_timestamp` (`timestamp`);
ALTER TABLE `player` ADD INDEX `player_idx_player_id` (`player_id`);

The optimized query:

SELECT
        a.player_id,
        COUNT(a.player_id) AS views,
        (SELECT
            b.firstname 
        FROM
            player AS b 
        WHERE
            (
                a.player_id = b.player_id
            ) LIMIT 1) AS firstname,
        (SELECT
            b.lastname 
        FROM
            player AS b 
        WHERE
            (
                a.player_id = b.player_id
            ) LIMIT 1) AS lastname,
        (SELECT
            b.link_id 
        FROM
            player AS b 
        WHERE
            (
                a.player_id = b.player_id
            ) LIMIT 1) AS link_id 
    FROM
        buyout_calculator_query AS a 
    WHERE
        a.timestamp > 259200 
    ORDER BY
        views DESC