Funnily enough I find backgrounds a case of "close, but no cigar". I find them mechanically flawed because the incentive is 100% to have one big one and then try to shoehorn it into every conceivable situation. This in turn necessitates GM policing of the use of backgrounds, which I find retrograde.
To me this is a "feature" not a bug. I think you miss the point of backgrounds entirely because you're viewing 13th Age backgrounds through the traditional D&D lens as a game of strategically managing limited resources. Hence the misplaced concern that a player is somehow "getting away with something" they shouldn't if they create a broad encompassing background.
13th Age's approach is a completely different paradigm than the one established in prior editions of D&D, where your PC's chance of survival with a critical skill check was won or lost during the character creation mini-game. Essentially the player is forced to strategically anticipate what skills are more important during character creation. They are then punished for guessing "wrong" if they put ranks into Tumble when they should have put ranks into Swim and now they can't swim well enough to make their check, so they die. Or rewarded for guessing right, when they can make a difficult skill check because they anticipated the need for it correctly and put all their ranks into it.
But that sort of strategic management of skills is not what 13th Age cares about. Now 13th Age does have areas where management of resources does matter. But this element of the game has been moved solely to the combat side, where you manage your spells, and powers and whatnot.
The 13th Age approach to skills is different. Backgrounds are really a storytelling tool cleverly disguised as a skill system. The point of the game is create an interesting story through play, not to test the player's ability to anticipate the future importance of skill choices required now. If you have a broad background in a 13th Age game it doesn't matter. Its simply a means for the DM to engage with the player and make the story more compelling. Why does your PC's background as an Imperial soldier let him swim better?
Well, perhaps the player creates a fascinating tale of how he was assigned bodyguard duty for the young prince while still a raw recruit. The young adolescent prince and his playmates liked to swim in a lake near the castle, and occasionally the player was forced to pull one or two of them out of the water who swam too far out and panicked.
Cool, now the player has some deeper ties to the campaign, and some important NPCs, and added some flavor. So heck yeah, give them the +5 for that swim check now. Does that help them succeed on the check, sure? Why not? Because the entire purpose of the background system as a story enabler has been achieved. Its not about punishing the player for making a poor resource allocation choice in the form of skill ranks.
You can certainly play 13th Age in a more traditional D&D style, if you want, though. So, if you want backgrounds to matter as a more traditional risk vs reward resource management way, then yeah, you probably would want to more carefully define how broad or narrow a PC's background choices can be.