I would say, as far as the "why is it this way/can we do this weithout mucking things up" question, it is primarily a "flavor balance" thing. They put in a "sword & spell" guy, but they wanted or needed it to be distinct from a bard (who gets enchantments and illusions, amongst other things), the arcane trickster (who gets enchantment and illusion), and the blade warlock (who gets a smattering of everything). Makes sense, to reinforce the "fighter subclass" flavor, they would make the make all about "fighting", attacking and defending, i.e. evocation and abjuration (with a smattering of everything else for spells that might fall outside those schools and/or allow a limited utility). Of the EK's known spoells, only 2 of the 3 starting spells hav to be evo. or abj. When they gain a new spell or spell level, they have to choose evo. or abj. But, at every level up, they can swap out spells and there is no indication of limitation to the schools for that.
So, long observation short, it's really just about presenting a "subclass" they could put under fighter...a guy with "fighting magic"...but there's no real reason it has to stay that way.
As @
Piratecat mentions above, an Elvin EK who is all about illusion & charms is totally a doable archetype with the EK class. I could also see Jedi-esque fighters (which the class as a whole is probably there to foster): flippy acrobatic light-armored warriors with some enchantments (mind magic) and telekinetic-type spells (mage hand, expeditious retreat, jump, etc...) but nothing flashy 'splodey. The Dwarf kingdom's shock trooper "Fire-Knights of the Great Forge" (I'm thinking a Knights Templar-like order/cabal revering Moradin [instead of Baphomet], who specializes in and uses only spells of earth and fire, also known colloquially as the "Magmancers"

.
And, as is always my preferred attitude in D&D, if you see something you don't like or want to do differently, CHANGE IT! Specifically, in this case, I see no reason the system will somehow falter or "break" by doing what has been suggested (and just allowing any spell schools to EK PCs).
I don't see or think there is any actual "inter-class power level balance" that is going to be thrown into a tizzy.