People are drawn to competition. It doesn't really matter what it is, as long as it's something you can practice at and get better than someone else at, and it gets the right marketing and advertisement, people will start caring about who might win, or who's the best at whatever.
Esports will likely become a mass-market popular thing at one point, but I think the best way to introduce it to the masses would be through something like Quake 1v1 matches.
The versatility, finesse, etc - the range of the Quake mechanics, or even something like the Delta Force 1-2-3 mechanics, allows the skill roof to be quite high, or it allowed a lot of room for improvement and practice. Something like Call of Duty has a, comparatively, very low skill-roof - the range of mechanics in CoD means that difference between a great, experienced player and a beginner isn't nearly as significant as in Quake or DF games.
One of the definitions of any competitive sport should be that an experienced player can, in general, absolutely dominate a beginner - ie, it has enough variables and complexity to allow a lot of room for improvement.
But, then again, I think there's another reason why the general public might have a hard time accepting "eSports" as something legitimate - every game, the graphics, rules, mechanics, interaction changes. Not to mention that balance is entirely up to the developers through patches. It's completely up to the developers of the game, who aren't affiliated with the current eSports scene, or have any ties to it, not any outside, general & enforceable, rules.
In order for eSports to have more legitimacy, there'd need to be a separate agency that is capable of accepting or not accepting certain games into a competitive league (Also a separate corporation.), has the power to make the developers fine-tune their game to make it correspond to whatever the rules are for certain games.
Right now, eSports just goes after whatever game that comes out, and that people play.
To make an eSports scene like this happen, though, there'd need to be some incentive for developers to want to do this - once there's enough money in eSports, though, it's probably very likely that they will.
The most efficient and likely way something like this will play out, though, is probably similar to how NFL works, or whatever it's called - an agency/corporation, probably a game-publisher, will create their own "competitive" games, leagues, set their own rules, allow anyone to play and practice with their game. Maybe they'd hire their own pro teams and let them play against each other, or maybe they'd just arrange the competitions where unaffiliated teams play against each other - ie, Coca Cola team, sponsored by Coca Cola, plays against Pepsi team, sponsored by Pepsi. Or official school teams. Winner gets money from the maker of the game. That's quite similar to how it works nowadays I suppose.
But games are created and limited by technology, which means that there are a bunch of variables involved. Graphics, means of interaction, mechanics, balance and so forth. This means that a separate corporation that sets and enforces the rules, and a corporation that hosts the game would have to enter the picture, or something similar - ie, publisher submits game to the corporation that hosts games for professional competition, the rules corporation decides whether or not it obeys the rules of whatever game category they submitted it to, and makes sure any necessary patches do so too. Why, or how, some separate corporations would have the power, or even the incentive to do so, I do not know.
Maybe one of them allows the game to broadcast matches on TV (Ie, major network channel.), but in order to do so, the game would have to follow the rules of that other corporation (Ie, pro gaming league that determines whether or not that game can be played professionally.). All though, one corporation that creates the games, sets the rules and maintains the pro competitive scene, with the teams being sponsored and trained by unaffiliated corporations or agencies, is probably the easiest way.
Basically, there needs to be a "set" game, with set rules, like Chess, or Poker, that the audience can get a good grasp of. But is there even any incentive to do so when the technology allows for so much variety?