Most sport are
just games, too. Chess is also widely considered a sport and it is probably even less physically challenging than most video games.
I guess you can make
nearly everything in a competition and therefore a sport.
problem with it being a sport is that its very heavily dependent on the game, there is no universal game that people only play.
football everyone plays football. but in video games everyone plays a variety of different types of games all made by different companys.
This plays a role, too. Although in a different manner. Of course people think about football (and American Eggball), the Olympic Games etc. first when talking about Sports. But just because less popular sports like curling, polo or whatever do not come to the minds of people that quickly, this does not mean they are no sports.
Going by that logic, this kind of means that "video games" is no sport. Instead, Starcraft is a sport. Dota is a sport, LoL is a sport. Maybe even CoH2

There is no "champion of eSports" just like there is no "champion of sports". In fact, very few pro-athletes participate in other sports (e.g. Football and Handball or whatever). Even although some of them may be very similar (e.g. MOBAs), they are still their own sports.
To someone who does not know the rules, Rugby and American Football might sound very similar when explained to them as well.
Another aspect is complexity. When it comes to eSports, often it is argued that many of them aren't very approachable to the casual observer, too convoluted to understand etc.
But how many of you know all the rules and all aspects of most sports? People might understand that to win at Football, the ball needs to be kicked into the goal and they will still be excited and cheer when a goal is scored.
However, many of them will not understand the intricacies of formations, tactics, strategies, switching players etc. that are related to it and they probably will not enjoy watching the ball being passed around as much as someone who is very interested into said intricacies.
Likewise, casual observers of eSports may as well enjoy big flashy team battles in LoL or several tanks blowing up in CoH2. Likewise, most will not understand everything that goes on and likewise they may very well not enjoy certain aspects of the game (e.g. laning phase in LoL) they are watching whereas someone else may understand and enjoy more of it.
@CombatMuffin: I think that -sadly- the influence of some sports is already too high spread in some eSports. However, this is also the consequence of there either being no or not enough behavioral rules that are actually enforced in eSports. Many sports clubs/federations have penalties for shit-talking, insulting others etc. and others can even be described as having a high focus on discipline.
Doesn't seem to be the case
in some eSports.