As people have said, the WTF has the support of the South Korean government and is a more unifed body rather than the very fractious ITF and related associations. If you look at the sports in the Olympics there is usually only one internationally recognised body, which may oversee national federations.
Unfortunately ITF based associations have a history of falling out with each. I remember when instructing at a university club, we used to try and promote the British Student Taekwondo Federation and it's competitions. The WTF side never had any issues and the competitions ran pretty smoothly. On the ITF side, some associations refused to take part as it would mean recognising other associations or clubs. I just thought it was a shame as it was a genuine attempt to have a competition that all university students, regardless of association could take part in. There was also the issue that some referees refused to fairly mark forms where the competitor didn't do sine wave. Ultimately it looks like the BSTF is only going to viable going forward as a WTF competition.
I understand that the country hosting the Olympics is allowed to introduce demonstration sports and, as Lesley said, when the Olympics were in Seoul it was introduced and later become a full Olympic sport. I think there was a vote on whether to keep it for the last Olympics and it only stayed in by one vote, so who knows if it will remain in the games much longer.
I personally would like to either see some sort of sport
karate or kickboxing in the Olympics, but given the numerous associations and governing bodies, whether this will ever be possible is debatable.