Am I missing something that is supposed to balance this? Are there hidden features that Battle Master and Champion get that raise their damage to a similar range? Is it just acknowledged that Eldritch Knight is the king of fighter damage, and the other subclasses have something else to balance it with?
Yes you are missing something.
To use War Magic, you have to use a magic action, that means it is one cantrip.
So take your 20th level Eldritch Knight. With a Greatsword, a 20 strength and a 20 intelligence using Warmagic to cast green flame blade he is going to do 4d6+3d8+10 (37.5) to one target and 3d8+5 (19.5) to a second target, if there happens to be a second target within 5 feet.
The same EK attacking 4 times as an action will do 8d6+20 (48) to up to 4 targets, you don't have any conditions to that, as long as you can reach them and attack them you get that damage. So using the cantrip is an extra 9 points of damage if (and only if) there is another enemy in 5 feet and it uses the Eldritch Knight's bonus action to do it, and he had to boost intelligence to 20 to do that. Same with Booming Blade - you get the extra damage only if the enemy moves.
Also these are usable on a melee attack only from within 5 feet. No using ranged weapons, no using reach weapons.
If you are comparing to other fighter subclasses, a Battlemaster can add 1d12 plus a pretty big condition or effect on an attack 6 times a short rest (which is almost once a round).
A champion is going to be scoring a crit on an 18.
So yes they have subclass abilities that do more damage too.
War Magic is a nice buff on an EK from levels 7-10 and in that range they are probably the highest damage fighter, but not so much at other levels.
What really makes the EK so strong is not the blade cantrips, it is the shield spell.