Why would you go great sword on an EK? greataxe or falchion are both better.
Greatsword is superior to both of those weapons in terms of raw numbers. 8.4/7.5/6.5.
There are additional factors for both comparisons also.
Greatsword has a smoother animation than Great Axe and when moving it does even more damage because of the animation.
Greatsword has better reach than Falchion and more reliably engages multiple targets as you sweep by.
I've been playing this type of build solo for 5+ years, occasionally switching to a caster-centric build, either PM or Bard. In that time I have tried every other reasonable weapon including Staff, Great Axe, Falchion, Hand Axe, Bastard Sword, Dwarven Axe, Repeating Heavy Xbow (don't ask) and Great Xbow (ditto).
Of all the weapons the ones that are most successful at THF are Greatswords because the baseline is just better, the animation is butter smooth and there is a much better selection of named weapons across the leveling arc.
The ones most successful at SWF are Daggers because Vistani Knife Fighter.