According to Wikipedia, the 2020 Summer Olympics will feature karate, and it will include Kata. However, tae kwon do has been an Olympic sport since the 1980s and it obviously does not include Poomse, even though there are major WTF-sanctioned Poomse competitions. Why are Karate Kata included at the Olympics but not Tae Kwon Do Poomsae?
There are several reasons, but in my opinion, they are somewhat of a cop-out - and yet a relief. Keep in mind, getting into the Olympics is a herculean task: there needs to be a federation representing 75 countries on 4 continents (for men) and 40 countries on 3 continents (for women), anti-doping policies, "modern appeal", media interest, promote gender equality, and a few other criteria.
When Taekwondo first came into the Olympic spotlight, they needed to showcase something the IOC could understand. Most people inside and outside of martial arts have a concept of sparring, so, that was sort of a given. But few people outside martial arts understand poomsae/kata. They may have seen it, but they surely don't understand the technical side of it: what they are, what they do, and what goes into judging a performance. And in the case of poomsase, specifically, they're very short compared to their Japanese kata bretheren - Koryo is only 32 movements. You blink, you miss half the performance. That leaves lay people thinking, "what was that all about??" This all goes to the youth appeal that the IOC wants, as well as media popularity. You ever see ESPN or ESPN2 showcase kata or poomsae and the world championship events? Not often. So this is an uphill and carefully scripted battle that the Karate (and TKD) federations and organizers will have to negotiate with the IOC.
Fast forward to 2012 with the scandals in sparring, specifically, Angel Matos' referee kick. Although he was kicked out of the Olympics and banned from the WTF for life, the sport, too, was placed on notice: fix it so it doesn't happen again, or you're gone. That was no time to be introducing "new" elements to the sport. As it stands now, the sport has settled quite a bit - it is actually a model for many other sports.
But Karate was also vying for a spot on the podiums as well. And there's Olympic committee rules, too. Among many rules, including a few subjective ones, they won't consider a sport if it is deemed too similar to another. For Karate to enter the stage with only sparring, it could appear too similar to Taekwondo.
(Note that in 2015 when Karate placed a bid to enter the Olympics, Softball and Baseball did the same. The latter two were also granted a spot, but they are considered one sport. Few men play softball, and few women play baseball, so you can imagine the bruhaha resulting if either - but not both - were included!)
So for this reason, Karate had to distinguish itself from Taekwondo - an irony, I know - and so, decided to include kata. It isn't yet determined whether a competitor may compete in either or both events, or that they must compete in both events, in order to receive a medal. My guess is that they'll have to compete in both; otherwise, it might look like kata was unceremoniously jammed in there to make itself distinguished from TKD.
Karate still has a long way to go to explain itself to the general public - and the IOC - about what kata is. However, many in TKD have no idea what poomsae is and what its purpose is; if Karate plays its cards right, those in TKD will get schooled in that concept. In other words, Karate's marketing can easily benefit the TKD followers, even though TKD competitors won't be competing in poomsae in the Olympics. However, TKD did remove a great deal of subjectivity with regards to scoring sparring with the use of electronic scoring. Karate will need to do the same, or it will have a hard time staying on the podiums. But with sparring, it was easier to have e-scoring - this can't be said of kata. There must be humans judging the performance, and this is going to be Karate's downfall if the Karate committee doesn't do this right.
Keep in mind, when "something" is included in the Olympics, it can be a sport, a discipline, or an event. For instance, triathlon was admitted as a sport at the 2000 games. Women’s wrestling was a new discipline in the sport of wrestling at Athens, and women’s pole vaulting debuted in Sydney as a track-and-field event. Karate is wholly another sport. Kata/poomsae would be considered an event, and it is likely that Karate would be considered two events: sparring and kata. What it has actually done, I don't know.
I think poomsae will eventually be added, but only once it can do so by offering objectivity. They'll surely watch Karate to see how they fare.