A word is merely a container of similar things, such that people can communicate more efficiently.
A word's meaning can shift over time and sometimes we create new words to communicate better.
The word 'sports' gives the readers/listeners a similar visualization. Right now, the majority of the population would not think of video games when we hear/read the word 'sport'. So right now - the answer is no.
However eventually, esport may become a dominate form of 'competitive entertainment' at which time current sports may be seeing a reverse decline. If such a scenario ever comes, we may say 'hey league is so common now, it is definition of sport'.