I don't have a Russian trauma as many people seem to have, although I've played a lot of Dota 2 already. Hmm.
Anyway, Kasparov is right by saying that elegant game design like Chess and Go will not disappear. But the forms in which it is delivered change as well (real boards with different figure styles, chess video games, gaming over the net), if we wanna draw an analogy to the graphics of games.
I btw don't think that there are no video games that will not played in hundreds of years. Graphics or input devices might change over the years, but it is likely that there will still be games like Bejeweled and Tetris.
How will competitive video games fare... I don't know. Depends on the following of people and we can only really estimate it once an eSport grows to a level where parents actually introduce their kids to the game and this will take years and a long life time of the same game principle. In the end we already have reoccurring themes, like RTS, Arena Shooter, ARTS, Racing Games, Shmups, Fighting Games, ...... which are the same in their quintessence (most of the time ^^). It is not a huge step to just keeping the gameplay of one "proven to be great" game and just keep up to date with the tech environment. It is happening with Fifa since 1993 (although I don't like the marketing strategy behind it), it is happening with Dota since 2005 (or earlier, if you want to count the Guinsoo era, although it was a lot less thought through then), ... .