It's kind of funny that US Olympic success is largely funded by people sitting on couches.
Wrestling, boxing, taekwondo, and judo are all martial arts officially contested during the 2016 international games. Additionally, Budo (Japanese) was exhibited in 1964, Glima (Viking) in 1912, and Savate (French) in 1924.
Demonstration sports were suspended after 1992, so Wushu (Chinese) was not exhibited at the Beijing games. Instead, a Wushu tournament ran concurrently with the international games, in Beijing, with permission from the international committee. Looked like a duck; quacked like a duck.
Sports with a compliant international governing body now get to be in a differently-branded international games every 4 years rather than a one-time demonstration sport just when it is popular in the current host country. Sports can be "promoted" from the "next year" games to the main event. Ju-jutsu, Sumo, Muay Thai, Kickboxing, and Karate will be contested in 2017. I am not familiar with any reason why the International Wushu body chooses not to organize events at the other games.
But wrestling is not removed. Wrestling is in for 2020, as is Karate--receiving one of the aforementioned promotions.